The point of intersection of the equations C = 100t + 5000 and C = 75t + 7000 represents the the point in time where the cost of installing either will be the same.
What is Linear Equations?Linear equations are equations where the right hand side and left hand side involves expressions with one or more variables for which the highest degree of the variable being 1.
We have two linear equations given,
C = 100t + 5000 and C = 75t + 7000.
The point of intersection of these linear equations will be when both the equations are equal.
100t + 5000 = 75t + 7000
100t - 75t = 7000 - 5000
25t = 2000
t = 2000/25
t = 80
This is the point in time where the cost of installing either will be the same, that is after 80 months.
Cost of installing = (100 × 80 ) + 5000 = 13,000
Or = (75 × 80) + 7000 = 13000
Hence the point of intersection of these equations represent the point in time where the cost of installing either will be the same.
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f a ball is thrown in the air with a velocity 52 ft/s, its height in feet t seconds later is given by y = 52t − 16t2.
(a) Find the average velocity for the time period beginning when t = 2 and lasting
(i) 0.5 second
(ii) 0.1 second.
(iii) 0.05 second.
(iv) 0.01 second.
(b) Estimate the instantaneous velocity when t = 2.
If a ball is thrown in the air with a velocity of 52 ft/s, its height in feet t seconds later is given by y = 52t − 16t2.
Then the average velocity for the time period beginning when t = 2 and lasting for 0.5s,0.1s,0.05s,0.01s will be -20,-13.6,-12.8,-12.16 respectively
y=y(t)=52t-16t²
At t=2, y=52(2)-16(2)² =40
The average velocity between time 2 and 2+h is Vav. = [tex]\frac{y(2+h)-y(2)}{(2+h)-2}[/tex]
=[tex]\frac{-12h-16h^2}{h}[/tex]
= -12-16h
Now finding the average velocity for the period beginning when t = 3 and lasting-
(i)0.5 s is (2,2.5) , Vav. = -20.0 ft/s
(ii)0.1 s is (2,2.1), Vav = -13.6 ft/s
(iii) 0.05s is (2,2.05), Vav. = -12.8ft/s
(iv) 0.01s is (2,2.01) Vav.=-12.16ft/s
And at t=2 , h will approaches to zero so Vav= -12ft/s
Therefore, If a ball is thrown in the air with a velocity of 52 ft/s, its height in feet t seconds later is given by y = 52t − 16t2.
Then the average velocity for the time period beginning when t = 2 and lasting for 0.5s,0.1s,0.05s,0.01s will be -20,-13.6,-12.8,-12.16 respectively.
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Researchers in the Hopkins Forest (see Problem 7-16 from) also count the number of maple trees (genus
acer) in plots throughout the forest. The following is a histogram of the number of live maples in 1002 plots
sampled over the past 20 years. The average number of maples per plot was 19.86 trees with a standard
deviation of 23.65 trees.
a). If we took the mean of a sample of eight plots, what would be the standard error of the
mean?
b). Using the central limit theorem, what is the probability that the mean of the eight would be
within 1 standard error of the mean?
c). Why might you think that the probability that you calculated in (b) might not be very
accurate?
a) The standard error of the mean is equal to the standard deviation divided by the square root of the sample size. In this case, the standard error of the mean would be equal to 23.65/sqrt(8) = 6.61.
b) Using the central limit theorem, we can say that the probability that the mean of the eight plots would be within 1 standard error of the mean is approximately equal to 68%. This is because, under the central limit theorem, the distribution of the sample means approaches a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size. Therefore, in this case, we would expect that 68% of the sample means would be within 1 standard deviation of the population mean.
c) The probability that we calculated in (b) might not be very accurate because the central limit theorem assumes that the underlying population is normally distributed. However, it is not clear from the histogram whether the number of live maples per plot is normally distributed. If the distribution of the number of live maples per plot is not normal, then the central limit theorem would not be applicable and the probability we calculated might not be accurate.
Hence we get the required answer.
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A basketball manufacturer selects a random sample of its basketballs each week to ensure a consistent air pressure within them is maintained. In Week 1, the sample had a mean air pressure of 8.2 psi (pounds per square inch) and a margin of error of 0.1 psi. In Week 2, the sample had a mean air pressure of 7.7 psi and a margin of error of 0.3 psi. Based on these results, which of the following is a reasonable conclusion? A) Most of the basketballs produced in Week 1 had an air pressure under 8.2 psl, whereas most of the basketballs produced in Week 2 had an air pressure under 7.7 psi. B) The mean air pressure of all the basketballs produced in Week I was 0.5 psi more than the mean air pressure of all the basketballs produced in Week 2 C) The number of basketballs in the Week 1 sample was more than the number of basketballs in the Week 2 sample. D) It is very likely that the mean air pressure of all the basketballs produced in Week 1 was less than the mean air pressure of all the basketballs produced in Week 2.
B) The mean air pressure of all the basketballs produced in Week I was 0.5 psi more than the mean air pressure of all the basketballs produced in Week 2
What is manufacturer?
A manufacturer is a person or business that turns raw materials into completed products using a variety of tools, machinery, and procedures, and then sells those products to customers, wholesalers, distributors, retailers, or other manufacturers who can use them to make more complicated products.
A basketball manufacturer selects a random sample of its basketballs each week to ensure a consistent air pressure within them is maintained. In Week 1, the sample had a mean air pressure of 8.2 psi (pounds per square inch) and a margin of error of 0.1 psi. In Week 2, the sample had a mean air pressure of 7.7 psi and a margin of error of 0.3 psi. Based on these results, Option B, which states that the average air pressure of all basketballs created during week one was 0.5 psi higher than that of all basketballs produced during week two, is the right response.
Option C is not discussed in any detail.
Factually, Option D is inaccurate.
Because the margin of error would have been 0, Option A is wrong.
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give the component from the following vectors shown
determine what type of statistical inference should be used to answer each of the questions below. (you will only use each response one time.)
type of statistical inference should be used to answer each of the questions below.Two-sample t-test is there a difference in the amount of time people spend on social media between genders.
Two-sample t-test is a type of inferential statistic used to determine whether there is a significant difference between the means of two independent groups.
In this case, the two groups would be genders (male and female) and the type of data is the amount of time spent on social media.
Two-sample t-test is there a difference in the amount of time people spend on social media between genders.
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Water is steadily dripping from a faucet into a bowl. You want to write an equation that represents the number of milliliters y of water in the bowl after . What is the constant of proportionality for the relationship between the number of milliliters y of water in the bowl and the time in seconds x? What is the equation?
Answer:
Read below
Step-by-step explanation:
The equation for the volume of water in the bowl after a certain amount of time can be represented by the formula: y = k * x, where k is the constant of proportionality. This constant represents the rate at which water is dripping into the bowl, in milliliters per second.
To find the constant of proportionality, you need to know the rate at which the water is dripping into the bowl. If you know the rate, you can plug it into the formula above to find the equation that represents the volume of water in the bowl over time. For example, if the water is dripping into the bowl at a rate of 1 milliliter per second, the equation would be: y = 1 * x. This means that after 1 second, the bowl would contain 1 milliliter of water, after 2 seconds it would contain 2 milliliters of water, and so on.
It's important to note that the constant of proportionality will change if the rate at which the water is dripping into the bowl changes. For example, if the water is dripping into the bowl at a rate of 2 milliliters per second, the equation would be: y = 2 * x. This means that after 1 second, the bowl would contain 2 milliliters of water, after 2 seconds it would contain 4 milliliters of water, and so on.
In summary, the equation for the volume of water in the bowl after a certain amount of time can be represented by the formula: y = k * x, where k is the constant of proportionality, which represents the rate at which water is dripping into the bowl. To find the equation, you need to know the rate at which the water is dripping into the bowl, and plug that value into the formula as the value of k.
Find the missing values of the ratio table
The missing values for the equivalent ratios in the order that they appear in this table include the following;
3/23/210What is a proportion?In Mathematics, a proportion can be defined as an expression which is typically used to represent (indicate) the equality of two (2) ratios. This ultimately implies that, proportions can be used to establish that two (2) ratios are equivalent and solve for all unknown quantities.
From column 1, we have:
Let the missing value be represented by x.
1/4/(1/2) = 3/4/x
2/4 = 3/4x
8x = 12
x = 12/8
x = 3/2
From column 2, we have:
Let the first missing value be represented by y.
1/4/(1/2) = y/3
2/4 = y/3
4y = 6
y = 6/4
y = 3/2.
From column 2, we have:
Let the second missing value be represented by z.
1/4/(1/2) = 5/z
2/4 = 5/z
2z = 20
z = 20/2
z = 10.
In conclusion, we can logically deduce that t 10/2, 5/(11/20), and 5/4.
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You are currently getting 26 sales opportunities per day and closing 64% of them.”I am closing ____ sales opportunities per day!”
I am closing 17 sales opportunities per day.
What is a percentage?The percentage is calculated by dividing the required value by the total value and multiplying by 100.
Example:
Required percentage value = a
total value = b
Percentage = a/b x 100
Example:
50% = 50/100 = 1/2
25% = 25/100 = 1/4
20% = 20/100 = 1/5
10% = 10/100 = 1/10
We have,
The total number of sales = 26.
64% of sales are closing.
This means,
The number of closing sales.
= 64% of 26
= 64/100 x 26
= 0.64 x 26
= 16.64
Thus,
The number of closing sales is 17.
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Gretchen spends 5.1 hours on Monday, 3/3/5 hours on Tuesday, and 5/3/10 hours on Wednesday studying for her final exams. If she spends 3/1/2 hours studying for each exam, how many exams does she have to take?
The number of exams that Gretchen has to take is 4 exams.
How many exams does Gretchen have to take?From the information illustrated, Gretchen spends 5.1 hours on Monday, 3/3/5 hours on Tuesday, and 5/3/10 hours on Wednesday studying for her final exams. The total hours that she used will be:
= 5.1 + 3 3/5 + 5 3/10
= 5.1 + 3.6 + 5.3
= 14 hours.
Since she spends 3/1/2 hours studying for each exam, the number of exams that she has to take will be:
= Number of hours / Hours used for each exam
= 14 / 3 1/2
= 4 exams
The number of exams are 4.
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The ABC Corporation is considering opening an office in a new market area that
would allow it to increase its annual sales by $2.6 million. The cost of goods sold is
estimated to be 40 percent of sales, and corporate overhead would increase by
$306,500, not including the cost of either acquiring or leasing office space. The
corporation will have to invest $2.6 million in office furniture, office equipment, and
other up-front costs associated with opening the new office before considering the
costs of owning or leasing the office space.
A small office building could be purchased for sole use by the corporation at a total
price of $5.8 million, of which $900,000 of the purchase price would represent land
value, and $4.9 million would represent building value. The cost of the building would
be depreciated over 39 years. The corporation is in a 21 percent tax bracket. An
investor is willing to purchase the same building and lease it to the corporation for
$645,000 per year for a term of 15 years, with the corporation paying all real estate
operating expenses (absolute net lease). Real estate operating expenses are
estimated to be 50 percent of the lease payments. Estimates are that the property
value will increase over the 15-year lease term for a sale price of $6.3 million at the
end of the 15 years. the property is purchased, it would be financed with an interest-
only mortgage for $3,280,000 at an interest rate of 4.5 percent with a balloon
payment due after 15 years.
Required:
a. What is the return from opening the office building under the assumption that it is
leased?
b. What is the return from opening the office building under the assumption that it is
owned?
c. What is the return on the incremental cash flow from owning versus leasing?
A:
The return from opening the office building under the assumption that it is leased would be the difference between the net income generated by the increase in sales and the expenses associated with opening the new office, including the cost of leasing the office space.
To calculate the net income generated by the increase in sales, we first need to calculate the total cost of goods sold. Since the cost of goods sold is estimated to be 40 percent of sales, the cost of goods sold for the new office would be 40 percent x $2.6 million = $<<40*.01*2.6=1.04>>1.04 million.
Subtracting the cost of goods sold from the increase in sales, we find that the net income generated by the increase in sales would be $2.6 million - $1.04 million = $<<2.6-1.04=1.56>>1.56 million.
Next, we need to calculate the expenses associated with opening the new office, including the cost of leasing the office space. Since the corporate overhead would increase by $306,500, the total up-front costs associated with opening the new office would be $306,500 + $2.6 million = $<<306.5+2.6=2.906>>2.906 million.
To calculate the cost of leasing the office space, we need to multiply the annual lease payment by the number of years in the lease term. Since the annual lease payment is $645,000 and the lease term is 15 years, the total cost of leasing the office space would be $645,000 x 15 years = $<<645000*15=9675000>>9.675 million.
Subtracting the total up-front costs and the cost of leasing the office space from the net income generated by the increase in sales, we find that the return from opening the office building under the assumption that it is leased would be $1.56 million - $2.906 million - $9.675 million = $<<1.56-2.906-9.675=-9.020>>-9.020 million. This means that the corporation would incur a loss of $9.020 million by opening the new office and leasing the office space.
B:
C:
To calculate the return on the incremental cash flow from owning versus leasing, we need to subtract the cash flow from leasing the office space from the cash flow from owning the office space.
First, let's calculate the cash flow from leasing the office space. We already know that the total cost of leasing the office space would be $9.675 million. To calculate the cash flow from leasing the office space, we need to subtract the real estate operating expenses from the total cost of leasing the office space. Since the real estate operating expenses are estimated to be 50 percent of the lease payments, the real estate operating expenses would be 50 percent x $645,000 = $<<50*.01645=322500>>322,500 per year. Since the lease term is 15 years, the total real estate operating expenses would be $322,500 x 15 years = $<<322.515=4837.5>>4,837,500. Subtracting the real estate operating expenses from the total cost of leasing the office space, we find that the cash flow from leasing the office space would be $9.675 million - $4.837.5 million = $<<9.675-4.837.5=4.837.5>>4,837,500.
Next, let's calculate the cash flow from owning the office space. We already know that the total up-front costs associated with opening the new office would be $2.906 million. To calculate the cash flow from owning the office space, we need to subtract the total annual operating expenses from the net income generated by the increase in sales. Since the net income generated by the increase in sales is $1.56 million and the total annual operating expenses are $217,774.36, the cash flow from owning the office space would be $1.56 million - $217,774.36 = $<<1.56-217774.36=1.34222564>>1,342,225.64.
Finally, to calculate the return on the incremental cash flow from owning versus leasing, we need to subtract the cash flow from leasing the office space from the cash flow from owning the office space. Therefore, the return on the incremental cash flow from owning versus leasing would be $1,342,225.64 - $4,837,500 = $<<1.34222564-4.837.5=-3.495.5>>-3,495,500. This means that the corporation would incur a loss of $3.495.5 million by choosing to own the office space instead of leasing it.
simplify (10x-5/6)-x+4/3
Answer:the simplified form of the expression (10x-5/6)-x+4/3 is 9x - 1.
Step-by-step explanation: To simplify the expression (10x-5/6)-x+4/3, you can start by applying the order of operations, which dictates that you should perform operations within parentheses or brackets first.
The expression (10x-5/6) can be simplified as follows:
(10x-5/6) = 10x - (5/6)
= 10x - (5*6)/6
= 10x - 5
= 10x - 5
Next, you can simplify the remaining terms of the expression:
(10x-5) - x + 4/3
= (10x-5) - (x3)/3 + 4/3
= (10x-5) - x + (43)/3
= (10x-5) - x + 4
= 10x - 5 - x + 4
= 10x - x - 5 + 4
= 9x - 1
Therefore, the simplified form of the expression (10x-5/6)-x+4/3 is 9x - 1.
MODELING REAL LIFE An obstacle course needs a
new piece made in the shape of a triangular prism
with an equilateral triangle for a base. The side length
x (in feet) of the base and the height y (in feet) of the
prism are related by the inequality y> x² + 1. The
piece has the following additional constraints.
• The height must be no more than 7 feet greater than
the side length of the base.
• The side length of the base must be at least 1 foot.
Write and graph a system that represents the situation.
Give one example of a height and side length that the
obstacle course can use.
The given inequality is y > x^2 + 1. To represent this inequality in a system of equations, we can subtract x^2 + 1 from both sides to get y - x^2 - 1 > 0. The inequality is nonstrict (i.e., it includes the equal sign), so we can represent it using an equation by setting y - x^2 - 1 = 0.
The additional constraints are y < x + 7 and x > 1. The first constraint can be represented by the inequality y - x - 7 < 0, which can be represented by the equation y - x - 7 = 0. The second constraint can be represented by the inequality x - 1 > 0, which can be represented by the equation x - 1 = 0.
Thus, the system of equations representing the situation is:
y - x^2 - 1 = 0
y - x - 7 = 0
x - 1 = 0
We can represent this system graphically by graphing each equation on a coordinate grid. The first equation, y - x^2 - 1 = 0, represents a parabola. The second equation, y - x - 7 = 0, represents a line with a slope of -1. The third equation, x - 1 = 0, represents a vertical line at x = 1.
One example of a height and side length that the obstacle course can use is a height of 6 feet and a side length of 2 feet. This satisfies all of the given constraints, and it can be seen in the graph as a point of intersection between the three lines.
we have the number of rinks resurfaced by zzz zambonis is equal to the rate for zzz zambonis times the time. we want the time for 1\,\text{rink}1rink1, start text, r, i, n, k, end text , and we want that time to be less than or equal to 5\,\text{minutes}5minutes5, start text, m, i, n, u, t, e, s, end text. therefore: 1\,\text{rink}
we want that time to be less than equation or equal to 5 minutes start text, m, i, n, u, t, e, s, end text. therefore: 1[tex]5minutes[/tex]
Rate for ZZZ Zambonis × Time ≤ 5 minutes
Time ≤ 5 minutes/Rate for ZZZ Zambonis
To solve for the time needed to resurface one rink, we first need to set up an equation. We know that the number of rinks resurfaced by ZZZ Zambonis is equal to the rate for ZZZ Zambonis multiplied by the time. We want the time for 1 rink to be less than or equal to 5 minutes, so we need to solve for time in the equation. We can do this by dividing both sides of the equation by the rate for ZZZ Zambonis, resulting in the equation Time ≤ 5 minutes/Rate for ZZZ Zambonis. This equation gives us the maximum time needed to resurface one rink.
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If the air temperature is -14F how many degrees must it rise to reach 0F
Answer: 14 degrees F.
Step-by-step explanation:
To see how many degrees it much rise to reach 0F, we will subtract -14 from 0 to find our answer.
o - - 14 = 0 + 14 = 14 degrees F
let sn be the number of successes in n independent bernoulli trials, where the probability of success for each trial is 1/2. evaluate the following limits.
(a) n→[infinity]lim P( 2n− 3n ≤S n ≤ 2n + 3n)
(b)n→[infinity] lim P( 2n− 4n ≤S n ≤ 2n+ 4n)
(c) n→[infinity]lim P(2n−20≤Sn ≤ 2n+20)
Let sn be the number of successes in n independent bernoulli trials, the probability of success for each trial is 1/2.
(a) 1 , (b) 0 ,(c) 0
(a) As n approaches infinity, the probability of having 2n - 3n successes in n trials is equal to the probability of having 2n + 3n successes in n trials. Since the probability of success for each trial is 1/2, the probability of having 2n - 3n successes is the same as the probability of having 2n + 3n successes. Therefore, the limit is 1.
(b) As n approaches infinity, the probability of having 2n - 4n successes in n trials is equal to 0, since the probability of success for each trial is 1/2. Similarly, the probability of having 2n + 4n successes in n trials is also equal to 0. Therefore, the limit is 0.
(c) As n approaches infinity, the probability of having 2n - 20 successes in n trials is equal to 0, since the probability of success for each trial is 1/2. Similarly, the probability of having 2n + 20 successes in n trials is also equal to 0. Therefore, the limit is 0.
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Write an equation of the line that passes through the given point and is (a) parallel and (b) perpendicular to the given line.
a. y= ?
b. y=?
The equation of line that passes through the given point and is (a) parallel to given line is 4x + y - 16 =0
(b) perpendicular to given line is 4x - y -8 =0
According to the question,
The required line is passing through given points and is
(a) parallel (b) perpendicular to given line
Let the equation of required line be (y - y') = m'(x - x')
(a) Parallel
If two lines are parallel to each other then their slopes are equal
Slope of given line : m = y2 - y1 / x2-x1
This line passes through (2,2) and (1,6)
=> m = 6 - 2 / 1 - 2
=> m = -4
Therefore , the slope of required line will be -4 also,
Required line passes through (4,3)
Equation of required line = (y - 4) = -4(x - 3)
=> y - 4 = -4x + 12
=> 4x + y - 16 =0
(b) Perpendicular
If two lines are perpendicular to each other then the product of their slope is -1
So, m × m' = -1
=> -4 × m' = -1
Dividing by -4
=> m' = 4
Therefore , equation of required line => (y - 4) = 4(x - 3)
=> y - 4 = 4x - 12
=> 4x - y -8 =0
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Addition Rule 03
The table below shows the soft drinks preferences of people in three age groups.
under 21 years of age
Cola- 40
Root beer- 25
Lemon lime- 20
between 21 and 40
Cola- 35
Root beer- 20
Lemon lime- 30
over 40 years of age
Cola- 20
Root beer- 30
Lemon lime- 35
If one of the 255 subjects is randomly selected, find the probability that the person is between 21 and 40 years of
age or that they drink root beer.
Enter your answer as a fully reduced fraction.
Probability provides information about the likelihood that something will happen.
What is probability explain?Probability is simply how likely something is to happen. Whenever we're unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. The analysis of events governed by probability is called statistics.It is based on the possible chances of something to happen. The theoretical probability is mainly based on the reasoning behind probability. For example, if a coin is tossed, the theoretical probability of getting a head will be ½.Probability is the branch of mathematics concerning the occurrence of a random event, and four main types of probability exist: classical, empirical, subjective and axiomatic. Probability is synonymous with possibility, so you could say it's the possibility that a particular event will happen.Meteorologists, for instance, use weather patterns to predict the probability of rain. In epidemiology, probability theory is used to understand the relationship between exposures and the risk of health effects.The calculated probabilities are
0.5455
0.5909
0.4615
0.5556
a. The probability of the people that prefer sprite is
Probability = 60/110
= 0.5455
B. The probability that a person is between 21 and 40
probability = 65/110
= 0.5909
C. Probability of drinking lemonade given that age is between 21 and 40
Probability = 30/65
= 0.4615
d. Probability of sprite when under 21
25/45
= 0.5556
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Please help will mark Brainly
The number of the adult will be 77
The number of the child will be 160.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
237 peoples used the public swimming pool.
And, The daily prices are $1.75 for children and $2 for adult.
Now,
Let The number of the adult = x
The number of the child = y
So, We can formulate;
237 peoples used the public swimming pool.
⇒ x + y = 237 ..(i)
And, The daily prices are $1.75 for children and $2 for adult.
⇒ 2x + 1.75y = 444 ..(ii)
From (i);
⇒ x + y = 237
⇒ x = 237 - y
Substitute above value in (ii), we get;
⇒ 2x + 1.75y = 444
⇒ 2 (237 - y) + 1.75y = 444
⇒ 474 - 2y + 1.75y = 444
⇒ 474 - 444 = 0.25y
⇒ 40 = 0.25y
⇒ y = 160
And, from (i),
⇒ x + y = 237
⇒ x + 160 = 237
⇒ x = 77
Thus, The number of the adult = 77
The number of the child = 160.
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the usher at a wedding asked each of the 808080 guests whether they were a friend of the bride or of the groom. here are the results
The probability that a randomly selected person from this sample was a friend of the bride OR of the groom is 0.9875.
Given that, at a wedding each guest asked the 80 guests whether they were a friend of the bride or of the groom.
What is the probability?
Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event.
Total Number of Guests which forms the Sample Space, n(S)=80
Let the event (a friend of the bride) =A
Let the event (a friend of the groom) =B
n(A) =59
n(B)=50
Friends of both bride and groom,
So,
n(A∪B)=n(A)+n(B)-n(A∩B)
n(A∪B)=59+50-30
n(A∪B)=79
The number of Guests who was a friend of the bride OR of the groom = 79
Thus,
P(A∪B)=n(A∩B)/n(S)
= 79/80
= 0.9875
Hence, the probability that a randomly selected person from this sample was a friend of the bride OR of the groom is 0.9875.
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Please help! (Attached image) remainder Theorem
The remainder when the polynomial is divided by x - 5 is given as follows:
B. 24.
The remainder when the polynomial is divided by x - 8 is given as follows:
B. 87.
What is the Remainder Theorem?The Remainder Theorem states that when a polynomial y = f(x) is divided by a binomial given by x - a, the remainder is given by f(a).
The polynomial for this problem is given as follows:
f(x) = 2x² - 5x - 1.
Hence the remainder when the polynomial is divided by x - 5 is calculated as follows:
f(5) = 2(5)² - 5(5) - 1 = 24. (option B).
(x - 5 = 0 -> x = 5 is the value at which the numeric value has to be calculated to obtain the remainder).
Hence the remainder when the polynomial is divided by x - 8 is calculated as follows:
f(8) = 2(8)² - 5(8) - 1 = 87. (option B).
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Show that if v1,...,vp is linearly dependent in V, then the set of images, Tv1,...,Tvp, is linearly dependent in W. This fact shows that if a linear transformation maps a set v1,...,vp onto a linearly independent set Tv1,...,Tvp, then the original set is linearly independent, too (because it cannot be linearly dependent.)
If v1,...,vp is linearly dependent in V, then there exist scalars c1,...,cp such that c1v1 + ... + cpvp = 0.
Applying the linear transformation T to both sides of this equation, we get T(c1v1 + ... + cpvp) = T(0). Since T is a linear transformation, T(c1v1 + ... + cpvp) = c1T(v1) + ... + cpT(vp).
Therefore, Tv1,...,Tvp is linearly dependent in W. This shows that if a linear transformation maps a set v1,...,vp onto a linearly independent set Tv1,...,Tvp, then the original set v1,...,vp must be linearly independent as well, because if it were linearly dependent, then Tv1,...,Tvp would also be linearly dependent.
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In the accompanying diagram of right triangle ABC,
altitude BD is drawn to hypotenuse AC.
Which statement must always be true?
1) AD/AB = BC/AC
2) AD/AB = AB/AC
3) BD/BC = AB/AD
4) AB/BC = BD/AC
The ratio of the length of the altitude to the length of one side of the triangle is equal to the ratio of the length of the hypotenuse to the length of the other side.1) AD/AB = BC/AC is true
In the accompanying diagram of right triangle ABC, altitude BD is drawn to hypotenuse AC. This means that the length of BD is equal to the length of the altitude from the vertex of the right angle to the hypotenuse. This also means that the length of BD is a right triangle's altitude. In a right triangle, the altitude of a right triangle is the line segment connecting one vertex of the right angle to the hypotenuse, and is perpendicular to the hypotenuse. This also means that the ratio of the length of the altitude to the length of one side of the triangle (AD/AB) is equal to the ratio of the length of the hypotenuse to the length of the other side (BC/AC). This statement must always be true, regardless of the size of the triangle or its specific shape. This is because the ratio of the length of the altitude to the length of one side of the triangle is always equal to the ratio of the length of the hypotenuse to the length of the other side.
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Marsha deposited $10,000 into a savings account five years ago. The simple interest rate is 4%. How much money did Marsha earn interest
The interest that Marsha will earn is $2000.
What will be the interest earned?Simply multiplying the daily interest rate by the principal and the number of days between payments yields simple interest. Consumers who pay their loans on time or early each month benefit from simple interest. Auto loans and short-term personal loans are typically interest-only loans.
In this case, Marsha deposited $10,000 into a savings account five years ago. The simple interest rate is 4%.
The interest will be:
= PRT
P = Principal = $10000
R = Rate = 4%
T = Time = 5
Interest = PRT
= $10000 × 4% × 5
= $2000
The interest is $2000.
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The following table shows the daily receipts in millions of dollars of the movie "Avatar" for successive Fridays after its opening on Friday 18 December 2009. Weeks 1 4 7 10 13 16 19 22 25 28 $Receipts 75.617 42.785 22.85 13.655 4.027 0.844 0.633 0.188 0.064 0.028 Estimate the instantaneous rate of change of daily receipts 13 weeks after the opening day.
The instantaneous rate of change of daily receipts 13 weeks after the opening day is 1.
To find the instantaneous rate of change of daily receipts 13 weeks after the opening day, you can use the definition of the derivative. The derivative of a function at a point is defined as the limit of the average rate of change of the function over a small interval around that point, as the interval becomes smaller and smaller.
In this case, the function we are interested in is the function that represents the daily receipts of the movie "Avatar" as a function of the number of weeks after the opening day. Let's call this function f(x), where x represents the number of weeks after the opening day.
To find the derivative of f(x) at the point x = 13, we can use the following formula:
f'(x) = lim(h->0) [f(x+h) - f(x)]/h
Plugging in the values, we get:
f'(13) = lim(h->0) [f(13+h) - f(13)]/h
Since we are given the values of the function at various points, we can plug in the values to find the derivative. In this case, we have:
f'(13) = lim(h->0) [(4.027 + h) - 4.027]/h
Simplifying this expression, we get:
f'(13) = lim(h->0) h/h
Since the limit of h/h as h approaches 0 is 1, we can say that:
f'(13) = 1
It's important to note that this result tells us the rate of change of the function at a single point in time, rather than over a longer interval. If we were interested in the average rate of change over a longer interval, we would need to use a different approach.
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Complete Question:
The following table shows the daily receipts in millions of dollars of the movie "Avatar" for successive Fridays after its opening on Friday 18 December 2009. Weeks $Receipts 1 75.617 4 42.785 7 22.85 10 13.655 13 4.027 16 0.844 19 0.633 22 0.188 25 0.064 28 0.028 Estimate the instantaneous rate of change of daily receipts 4 weeks after the opening day. Round to four decimal places. - 1.1726 TIP Enter your answer as an integer or decimal number. Examples: 3,-4, 5.5172 Enter DNE for Does Not Exist, oo for Infinity Get Help:
The height of a triangle is 8 em shorter than its base. If the area of the triangle is 24 cm2, find the height of the triangle a) 07cm b) O12 cm c) O 13 cm d) O 3cm e) O4 cmf) O None of the above
When the area is 24 cm², the height of the triangle is 4 cm.
What is area?The quantity area indicates the extent of a region on a planar or curved surface. Surface area refers to the area of an open surface or the border of a three-dimensional object, whereas area of a plane region or plane area refers to the area of a form or planar lamina. The area is the region defined by an object's form. The area of a form is the space covered by a figure or any two-dimensional geometric shape in a plane.
Here,
area=1/2(bh)
h=b-8
24=1/2(b(b-8))
48=b²-8b
b²-8b-48=0
b²-12b+4b-48=0
b(b-12)+4(b-12)=0
b=-4,12
b=12 cm
h=12-8
h=4 cm
The height of triangle is 4 cm when area is 24 cm².
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Let X be a binomial random variable with n trials and probability p of success. A.What is the generalized likelihood ratio for testing H_0: p = .5 versus H_A: p notequalto .5? B.Show that the test rejects for large values of |X - n/2|. C.Using the null distribution of X, show how the significance level corresponding to a rejection region |X - n/2| > k can be determined. D.If n = 10 and k = 2, what is the significance level of the test? E.Use the normal approximation to the binomial distribution to find the significance level if n = 100 and k = 10.
A: p not equal to the value will be [tex]$$N=\left(\frac{n / 2}{x}\right)^x\left(\frac{n / 2}{n-x}\right)^{n-9} .$$[/tex]
B.Using the null distribution of X, show how the significance level corresponding to a rejection region |X - n/2| > k can be determined.
C. If n = 10 and k = 2, the significance level of the test = 0.11
D. Use the normal approximation to the binomial distribution of the significance level if n = 100 and k = 10 the value will be N(0.1).
1. The value of the generalized likelihood ratio test statistic will be
[tex]A=\frac{\left(\begin{array}{l}n \\x\end{array}\right)(1 / 2)^n}{m_0 x_{0 < p < 1}\left(\begin{array}{l}n \\x\end{array}\right) p^n(1-p)^{n-x}}[/tex]
As we know, the denominate will be maximized when [tex]$p=x / n$[/tex], the P under [tex]\Omega[/tex]
Hence [tex]$$N=\left(\frac{n / 2}{x}\right)^x\left(\frac{n / 2}{n-x}\right)^{n-9} .$$[/tex]
b)
[tex]$$\begin{aligned}\Lambda & =\left(\frac{n / 2}{x}\right)^x\left(\frac{n / 2}{n-x}\right)^{n-x} \\& =\left[\left(\begin{array}{c}2 x \\n\end{array}\right)^{-2 x / n}\left(2-\frac{2 x}{n}\right)^{-2+2 x / n}\right]^{n / 2} \\& =\left[y^{-y}(2-y)^{-(2-y)}\right]^{n / 2} \\& =\left[(1+v)^{-(1+v)}(1-v)-(1-v)\right]^{n / 2}\end{aligned}$$[/tex]
where [tex]y=2 x / n=1+0[/tex]
The last expression is symmetric about [tex]\theta=0+[/tex] is a maximum [tex]$\rightarrow t v=0$[/tex].
since the test rejects [tex]H_0[/tex] when [tex]$\wedge$[/tex] it is small.
If the only if [tex]$|0|=|1,2 \pi| n \mid$[/tex] is large.
This is equivalent to |x-n/2| is large
c) The significance level [tex]$\alpha$[/tex] of the test that rejects H_0 when [tex]$|x-n| 2 \mid > *$[/tex] is given by [tex]$$P\left(|x-n / 2| > k \mid H_0\right) \text {. }$$[/tex]
This probability can be determined using the pmf of the B(n, 0.5) distribution.
d) If n=10+k=2,
[tex]$$\begin{aligned}& P\left(|x-n|=| > k| H_0\right) \\= & P\left(|x-5| > 2 \mid H_0\right) \\= & P\left(x=0 \mid H_0\right)+P\left(x=1 / H_0\right)+P\left(x=2 \mid H_0\right)+P\left(x=8 / H_0\right) \\& +P\left(x=9 \mid H_0\right)+P\left(x=10\left(H_0\right)\right. \\= & 0.11 .\end{aligned}$$[/tex]
e) Using the normal approximation to the binomial distribution B(100,0.5), x is approximately n amply distributed with mean 50 * variance n p(1-p)=25.
Thus, the significant level of the test is
[tex]$$\begin{aligned}& P(|x-50| > 10) \approx P(|z| > 2)=0.046 . \\& \text { where } z \sim N(0,1) \text {. } \\&\end{aligned}$$[/tex]
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Which of the following is two dimensional and infinitely large? A.a solid
B.a sphere
C.space
D.a plane
use lagrange multipliers to find the maximum or minimum values of the function subject to the given constraint. (if an answer does not exist, enter dne.) f(x, y, z)
The minimum and the maximum values are 0 and 1/27, respectively.
The extreme values of a function are the minimum and the maximum values of the function.
The function is given as:
f(x, y, z) = x²y²z²
x² + y² + z² = 1
Subtract 1 from both sides of x² + y² + z² – 1
x² + y² + z²-1=0
Using Lagrange multiplies, we have:
L(xy.z. λ) f(x. y.z) + λ(0)
Substitute f(x.y. z) = x³y²z² and x² + y² + z² − 1 = 0
L(x, y, λ) = x²y²z² + λ(x² + y² + 2² – 1)
Differentiate
Lx = 2xy²z² + 2λx
Ly = 2x²yz² + 2λy
Lz = 2x²y³z + 2λz
Lλ = x² + y² + z²
Equate to 0
2xy²z² + 2λx = 0
2x²yz² + 2λy = 0
2x²y²z+ 2λz = 0
x² + y² + z²-1=0
Factorize the above expressions
2xy²z²+2λx = 0
2x(y²z² + λ) = 0
2x = 0 or y²z²+λ=0
x=0 or y²z² = -λ
2x²yz² + 2λy = 0
2y(x²z² + λ) = 0
2y=0 or x²z²+λ=0
y = 0 or x²z² = -λ
2x²y²z+λz = 0
2z(x²y² + λ) = 0
2z = 0 or x²y² +λ=0
z = 0 or x²y² = -λ
So, we have:
x = 0 or y²z² = -λ
y=0 or x²z² = -λ
z = 0 or x²y² = -λ
Because
-λ= -λ
The above expressions become:
x=y=z=0
x²z² = y²z²
x² = y²
x = ±y
Also, we have:
x²y² = x²z²
y² = z²
y = tz
Also:
y²z² = x²y²
z² = x²
z = ±x
So, we have:
x = ±y
y = ±z
z = ±x
This means that:
x=y=z
Recall that: x² + y² + z² = 1
So, we have:
x² + x² + x² = 1
3x² = 1
Divide through by 3
x² = 1/3
Take square roots of both sides
x=±[tex]\frac{1} \sqrt{3}[/tex]
So, we have:
x=y=z=±[tex]\frac{1} \sqrt{3}[/tex]
So, the critical points are:
x=y=z=0
x=y=z=±[tex]\frac{1} \sqrt{3}[/tex]
Substitute the above values in f(x, y, z)=x²y²z²
f(0,0,0) = 0² × 0² × 0² = 0
f(±[tex]\frac{1} \sqrt{3}[/tex],±[tex]\frac{1} \sqrt{3}[/tex],±[tex]\frac{1} \sqrt{3}[/tex]) = (±±[tex]\frac{1} \sqrt{3}[/tex])² × (±±[tex]\frac{1} \sqrt{3}[/tex])² × (±±[tex]\frac{1} \sqrt{3}[/tex])² =[tex]\frac{1}{27}[/tex]
Considering the above values, we have:
Minimum 0
Maximum = 1/27
Hence, the minimum and the maximum values are 0 and 1/27, respectively.
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The table shows values for a quadratic function.
A. 32
X
0
1
2
3
4
5
6
B. 10
y.
0
What is the average rate of change for this function for the interval from x = 3
to x = 5?
2
8
18
32
50
72
Answer:
c
Step-by-step explanation:
Rewrite the set s by listing its elements. Make sure to use the appropriate set
notation.
S={x|x is an integer and -5
S=
The set S is composed of the integers that are between -5 and -1; with -5 being included. Rewriting this set by using the Roster Method gives us an answer of,
S = {-5, -4, -3, -2}
What is meant by roster method?The term "roster method" refers to a method of displaying a set's components by listing them in bracketed lists. The set can be modeled most easily using the roster form. The components of a set are listed in row format in roster form, with commas between them.
The components of a set are listed in row format in roster form, with commas between them.
A set in mathematical form is simply represented mathematically using the roster notation. The elements (or members) in this approach are listed in a row between curly brackets. When there are multiple elements in a set, there is a comma between every pair of elements.
Based on the given above, the set T is composed of the integers that are between -5 and -1; with -5 being included. Rewriting this set by using the Roster Method gives us an answer of,
S = {-5, -4, -3, -2}
The complete question is:
Rewrite the set S by listing its elements. Make sure to use the appropriate set notation. S = {[ z| z is an integer and -5 ≤ x < -1}
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