Answers
Answer:
[tex]\log_{10}\sqrt{a} -\dfrac{3}{4}=\dfrac{1}{2}\log_{10}a-\dfrac{3}{4}[/tex]
Step-by-step explanation:
Given:
[tex]x=\dfrac{\sqrt{10a\sqrt{0.1}}}{10}[/tex]
Therefore:
[tex]\implies \log_{10}x=\log_{10}\left(\dfrac{\sqrt{10a\sqrt{0.1}}}{10} \right)[/tex]
[tex]\textsf{Apply the log quotient law}: \quad \log_a \left(\dfrac{x}{y}\right)=\log_ax - \log_ay[/tex]
[tex]\implies \log_{10}\sqrt{10a\sqrt{0.1}}-\log_{10}10[/tex]
[tex]\textsf{Apply log law}: \quad \log_aa=1[/tex]
[tex]\implies \log_{10}\sqrt{10a\sqrt{0.1}}-1[/tex]
[tex]\textsf{Apply radical rule: $\sqrt{ab}=\sqrt{a}\sqrt{b}$, \quad assuming $a \geq 0,\;b\geq 0$}[/tex]
[tex]\implies \log_{10}\left(\sqrt{10}\sqrt{a}\sqrt{\sqrt{0.1}}}\right)-1[/tex]
[tex]\textsf{Apply the log product law}: \quad \log_axy=\log_ax + \log_ay[/tex]
[tex]\implies \log_{10}\sqrt{10} + \log_{10}\sqrt{a} + \log_{10}\sqrt{\sqrt{0.1}}}\right)-1[/tex]
Rewrite the radicals:
[tex]\implies \log_{10} \left(10\right)^{\frac{1}{2}} + \log_{10}\sqrt{a} + \log_{10}\left(0.1\right)^{\frac{1}{4}\right)-1[/tex]
[tex]\textsf{Apply the log power law}: \quad \log_ax^n=n\log_ax[/tex]
[tex]\implies \dfrac{1}{2}\log_{10} 10 + \log_{10}\sqrt{a} + \dfrac{1}{4}\log_{10}0.1-1[/tex]
[tex]\textsf{Apply log laws}: \quad \log_aa=1 \quad \textsf{and} \quad \log_a\dfrac{1}{a}=-1[/tex]
[tex]\implies \dfrac{1}{2}(1) + \log_{10}\sqrt{a} + \dfrac{1}{4}(-1)-1[/tex]
[tex]\implies \dfrac{1}{2}+ \log_{10}\sqrt{a} -\dfrac{1}{4}-1[/tex]
[tex]\implies \log_{10}\sqrt{a} -\dfrac{3}{4}[/tex]
If you want to simplify further:
[tex]\implies \log_{10}\left(a\right)^{\frac{1}{2}} -\dfrac{3}{4}[/tex]
[tex]\implies \dfrac{1}{2}\log_{10}a-\dfrac{3}{4}[/tex]
Answer:
56
Step-by-step explanation: 10a+10 0.1 -=
Related Questions
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.
Answers
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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Use the given information to prove that ΔRST ≅ ΔVUT.
Given: ST ≅ UT
∠SRT ≅ ∠UVT
Prove: ΔRST ≅ ΔVUT
Answers
Based on the given information , the proof of triangle RST ≅ triangle VUT is shown below .
In the question ,
two triangles are given where ;
it is given that side ST ≅ UT
and ∠SRT ≅ ∠UVT .
we have to prove that ΔRST congruent ΔVUT .
In triangle RST and triangle VUT .
side ST = side UT .....given that ST ≅ UT ;
angle SRT = angle UVT ....given that ∠SRT ≅ ∠UVT ;
angle RTS = angle VTU .....because they are vertically opposite angles .
Hence by ASA congruence , triangle RST ≅ triangle VUT .
Therefore , it is proved that triangle RST ≅ triangle VUT .
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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?
Answers
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.
give the component from the following vectors shown
Answers
v = 4 , 5
Mark me Brainly
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)
Answers
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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What improvement can we make?
Answers
In general, the improvements that we can make refer to optimizations that can be applied in a context.
¿What are the improvements?In a particular process or context, the best are optimizations that are done to make something more efficient or to improve the quality of an element.
For example, some improvements that we could make in our community are:
Establish more efficient containers to better collect garbage.Optimize public lighting. Improve solid waste management.¡Hope this helped!
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?
Answers
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.
differentiate of the following w.r.t.x
i mark u branlist if u give me right answer
Answers
The differentiation of y = logx - cosecx + 5ˣ + 3/(x^(3/2)) with respect to x is cotx.cosecx + log(5).5ˣ + 1/x - 9/(2x^(5/2))
How to differentiate y = logx - cosecx + 5ˣ + 3/(x^(3/2)) with respect to x?In mathematics, differentiation is the process of finding the derivative of a function. The derivative of a function is a measure of how the function changes as its input (or independent variable) change
Given: y = logx - cosecx + 5ˣ + 3/(x^(3/2))
Thus, the derivative will be:
d/dx [logx - cosecx + 5ˣ + 3/(x^(3/2))]
= d/dx [logx] - d/dx [logx] + d/dx [5ˣ] + 3d/dx[(x^(3/2))]
= 1/x - (-cotx)cosecx + log(5).5ˣ + 3(-3/2)x^((3/2) -1)
= cotx.cosecx + log(5).5ˣ + 1/x - 9/(2x^(5/2))
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determine what type of statistical inference should be used to answer each of the questions below. (you will only use each response one time.)
Answers
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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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=
Answers
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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Score on last attempt: 2.5 out of 3 Score in gradebook 2.5 out of 3 For each of the following accounts, determine the growth factor and percent change for the account value over one compounding period a Account A has a 5% APR compounded monthly (12 times per year). Determine the account value's growth factor and percent change per compounding period Preview 1. Monthly Growth Factor: 1.0041667 ii. Monthly Percent Change: (1.0041667-1)-100 % Preview b. Account B has a 6.9% APR compounded quarterly (4 times per year). Determine the account value's growth factor and percent change per compounding period. i. Quarterly Growth Factor: 14(0.069/4) Preview 11. Quarterly Percent Change: (1.01725-1)*100 % Preview c. Account Chas a 3.4% APR compounded daily (365 times per year). Determine the account value's growth factor and percent change per compounding period i. Daily Growth Factor: 1+(0.034/100) Preview 11. Daily Percent Change: (1.00034.1) 100 N% Preview Submit
Answers
given rate r = 5% compounded monthly r = 0.05/12 = 0.0041667
(1) monthly growth factor = 1.0041667 (2) monthly percent change = (1.0041667-1)*100%= 0.0041667*100% monthly percent change = 0.41667%
(B) given rate r = 6.9% compounded quarterly
r = 0.069/4 = 0.01725
(1) Quarterly growth factor = 1.01725
(2) Quarterly percent change = (1.01725-1)*100%= 0.01725*100% monthly percent change =1.725%
(C) given rate r = 3.4% compounded quarterly
r = 0.034/365 = 9.315 * 10 ^ - 5
(1) daily growth factor = 1 + 9.315 * 10 ^ - 5 = 1.00009315
(2) daily percent change = (1.00009315-1)*100%= 0.00009315*100% monthly percent change =0.009315%
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Please help will mark Brainly
Answers
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 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.
Answers
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:
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.
Answers
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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Cast Iron Grills, Incorporated, manufactures premium gas barbecue grills. The
company reports inventory and cost of goods sold based on calculations from a LIFO
periodic inventory system. Cast Iron's December 31, 2024, fiscal year-end inventory
consisted of the following (listed in chronological order of acquisition):
Units. Unit cost
6,800. 600
4,900. 700
7,800. 800
The replacement cost of the grills throughout 2025 was $900. Cast Iron sold 36,000
grills during 2025. The company's selling price is set at 200% of the current
replacement cost.
Required:
1. & 2. Compute the gross profit (sales minus cost of goods sold) and the gross profit
ratio for 2025 under two different assumptions. First, that Cast Iron purchased 37,000
units and, second, that Cast Iron purchased 19,500 units during the year.
4. Compute the gross profit (sales minus cost of goods sold) and the gross profit ratio
for 2025 assuming that Cast Iron purchased 37,000 units (as per the first assumption)
and 19,500 units (as per the second assumption) during the year and uses the FIFO
inventory cost method rather than the LIFO method.
Answers
Answer:
a) ending inventory: 11,850,000
cost of goods sold: 25,200,000
gross profit 25,200,000
b)
ending inventory: 1,800,000
cost of goods sold: 23,100,000
gross profit 50,400,000 - 23,100,000 = 27,300,000
5,200 at $600
4,100 at $700
6,200 at $800
purchase 29,000 at $900
-sold 28,000 grills
As we use LIFO we sale from the last purchase thus, 29,000 - 28,000 = 1,000 of this units are added as another layer for the inventory account
ending inventory
5,200 at $ 600
4,100 at $ 700
6,200 at $ 800
1,000 at $ 900
Total $ 11,850,000
cost of good sold:
28,000 x $900 = $25,200,000
sales revenue
28,000 x 900 x 200% = $50,400,000
gross profit sales revenue less COGS
b) 5,200 at $600
4,100 at $700
6,200 at $800
purchase 15,500 at $900
-sold 28,000 grills
we check how many layer deep we go:
28,000 - 15,500 at 900= 12,500
12,500 - 6,200 at 800= 6,300
6,300 - 4,100 at 700 = 2,200 at 600
Ending Inventory
3,000 at $600 = $ 1,800,000
COGS:
15,500 x 900 + 6,200 x 800 + 4,100 x 700 + 2,200 x 600 = 23,100,000
Step-by-step explanation:
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)
Answers
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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4 kilometer from house to the nearest mailbox. how far is it in meter?
Answers
Answer:
4 kilometers is equal to 4000 meters.
Step-by-step explanation:
1 kilometer is equal to 1000 meters, so 4 kilometers is equal to 4 x 1000 = 4000 meters. This is because the prefix "kilo" means 1000, so 1 kilometer is equivalent to 1000 meters.
4 kilometers is equal to 4000 meters.
- multiply the length value by 1000
You are currently getting 26 sales opportunities per day and closing 64% of them.”I am closing ____ sales opportunities per day!”
Answers
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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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}
Answers
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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Find the missing values of the ratio table
Answers
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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If w is FALSE, x is TRUE, and y is FALSE, what is ((w OR Y') AND (x' AND y')) OR ((w OR y') AND (x OR Y)') ? A) TRUE B) Not enough information.C) FALSE D) NULL
Answers
After solving the logical operations it can be determined that the correct answer is C) FALSE.
What are logical operations?Logical operations are phrases that connect expressions to test if they are true or not.
What does the complement sign mean?The complement (') sign means that the opposite value of the variable is needed. For example if y is false, y' means true.
Let False = 0 and True = 1
= ((0 OR 0') AND (1' AND 0')) OR ((0 OR 0') AND (1 OR 0)')
= ((0 OR 1) AND (0 AND 1)) OR ((0 OR 1) AND (1)')
= (1 AND 0) OR (1 AND 0)
= 0 OR 0
= 0
Hence, the answer is C) FALSE
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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.)
Answers
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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leg-leg (ll) congruence theorem if the legs of a right triangle are congruent to the legs of a second right triangle, then the triangles are congruent.
Answers
The leg-leg (LL) theorem of congruency states that if the legs of a right triangle are equal to the legs of a second right triangle, then the two triangles are congruent.
We know that two triangles are congruent if and only if they have three pairs of equal sides.
In the case of the LL congruence theorem,
the two triangles have two pairs of equal sides (the legs) and one pair of equal angles (the right angles).
Therefore by the Side-Angle-Side (SAS) congruence criterion, the 2 triangles are congruent.
Alternatively
If the legs are of equal length then
Their squares would be equal to
Hence, the sum of these squares too would be equal.
Now taking a root over these would give us the hypotenuse for these triangles, which in turn should be equal.
Hence, by SSS congruence criteria
the triangles would be congruent.
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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?
Answers
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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evaluate the integral below by interpreting it in terms of areas in the figure. the areas of the labeled regions are a1
Answers
The integral can be evaluated by interpreting it in terms of the area of the labeled regions in the figure. The integral is equal to the sum of the areas of the labeled regions, which is equal to a1.
1. The integral is given as an area under a graph.
2. The graph has labeled regions.
3. The integral can be evaluated by interpreting it in terms of the area of the labeled regions in the figure.
4. The integral is equal to the sum of the areas of the labeled regions, which is equal to a1.
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Which of the following sets of numbers could not represent the three sides of a right
triangle?
O {48, 55, 73}
O {39, 80, 89}
O {42, 56, 70}
O {10, 23, 26}
Submit Answer
Answers
{48, 55, 73} and {10, 23, 26} do not represent a right angle triangle.
What is the way of classifying a triangle?If a² + b² < c² then it is an acute angle triangle.
If a² + b² = c² then it is a right-angle triangle.
If a² + b² ≥ c² then it is an obtuse angle triangle.
Now,
48² + 53² = 73².
2304 + 2809 = 5329.
5113 ≠ 5329 so do not represents a right-angle triangle.
Similarly,
39² + 80² = 89² represents a right-angle triangle.
42² + 56² = 70² represents a right-angle triangle.
10² + 23² ≠ 26²so do not represent a right-angle triangle.
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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=?
Answers
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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At a time t hours after taking a tablet, the rate at which a drug is being eliminated is r(t) = 50(e^-0.1 t - e^-0.20 t) mg/hr. Assuming that all the drug is eventually eliminated, calculate the original dose. Enter the exact answer. The original dose was _______mg
Answers
At a time t hours after taking a tablet, the rate at which a drug is being eliminated is r(t) = 50(e^-0.1 t - e^-0.20 t) mg/hr. Assuming that all the drug is eventually eliminated, calculate the original dose. Enter the exact answer. The original dose was 0 mg.
A limit is a value that a function or sequence approaches as the input or index approaches a certain value. Limits are a fundamental concept in calculus, and they are used to define important ideas such as derivatives and integrals. They are also used in other areas of mathematics, such as real analysis and mathematical analysis.
The rate at which the drug is being eliminated is the derivative of the amount of drug present at time t. We can write this as:
r(t) = dA(t)/dt = 50(e^-0.1 t - e^-0.20 t) mg/hr
If we integrate both sides of this equation with respect to t, we get:
A(t) = ∫r(t) dt = ∫50(e^-0.1 t - e^-0.20 t) dt
This integral can be evaluated using integration by parts. Let u = e^-0.1 t and dv = e^-0.20 t. Then du = -0.1 e^-0.1 t and v = -1/(0.2) * e^-0.20 t = -5e^-0.20 t. We can then write:
A(t) = -5 ∫e^-0.1 t * e^-0.20 t dt + (1/0.2) ∫e^-0.1 t dt
The first integral on the right-hand side can be evaluated using the substitution y = -0.3t, which gives:
A(t) = -5 ∫e^y dy + (1/0.2) ∫e^-0.1 t dt
The first integral is simply -5e^y, and the second integral is -5e^-0.1 t. Substituting these results back into the expression for A(t) gives:
A(t) = -5(-5e^-0.3 t) + (1/0.2)(-5e^-0.1 t)
Combining constants and simplifying gives:
A(t) = 25e^-0.1 t - 25e^-0.3 t
We are told that all the drug is eventually eliminated, so A(t) approaches 0 as t approaches infinity. This means that the original dose A(0) must be equal to the limit of A(t) as t approaches 0.
Taking the limit as t approaches 0 gives:
A(0) = lim t → 0 [25e^-0.1 t - 25e^-0.3 t]
This limit is equal to 25 - 25 = 0, so the original dose was 0 mg.
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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?
Answers
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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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
Answers
Answer:
c
Step-by-step explanation: