As a result, if (0,0) is a solution to the system of inequalities above, the connection between a and b is a>b in the xy-plane.
What is inequality?In mathematics, an inequality is a connection between two expressions or values that are not equal to each other. Inequality originates from a lack of equilibrium. To solve an inequality, isolate the variable on one side from the other constants. To do so, use opposing procedures to modify the inequality. To begin, separate the x by multiplying each side by two. Whatever you do on one side must equally be done to the other.
Here,
Given, y<−x+a and y>x+b
As given, (0,0) is a solution to the above inequalities.
0<0+a and 0>0+b
a>0 and b<0
Thus, we can conclude from the above inequalities that a is a positive number and b is a negative quantity.
Therefore, it can be surely said that a>b.
Hence, the relationship between a and b is a>b in the xy-plane, if (0,0) is a solution to the system of inequalities above.
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assume that the two samples are independent simple random samples selected from normally distributed populations. do not assume that the population standard deviations are equal. refer to the accompanying data set. use a significance level to test the claim that women and men have the same mean diastolic blood pressure.
Women
Men
Women
Men
71.0
69.0
63.0
63.0
94.0
64.0
72.0
79.0
81.0
69.0
56.0
91.0
66.0
60.0
59.0
79.0
71.0
64.0
79.0
87.0
79.0
71.0
73.0
63.0
57.0
59.0
69.0
71.0
50.0
40.0
65.0
78.0
71.0
67.0
66.0
75.0
87.0
66.0
83.0
57.0
68.0
83.0
54.0
65.0
60.0
75.0
75.0
84.0
71.0
53.0
75.0
71.0
1. Let μ1 be the mean diastolic blood pressure for women and let μ2 be the mean diastolic blood pressure for men. What are the null and alternative hypotheses?
2. Calculate the test statistic.(Round to two decimal places as needed.)
3. Find the P-value. (Round to three decimal places as needed.)
4. Make a conclusion about the null hypothesis and a final conclusion that addresses the original claim.
1)test the claim that women and men have the same mean diastolic blood pressure .Let p1 be the mean diastolic blood pressure for women and let u2 be the mean diastolic blood pressure for men
there for we test
H0: u1 = u2
H1 : u1 =!u2
2)t= overline X 1 - overline X 2 sqrt((S_{1} ^ 2)/n_{1} + (S_{2} ^ 2)/n_{2})
= (69.81 - 69.35)/(sqrt((10.46 ^ 2)/26 + (11.2l ^ 2)/26)
= 0.153
3) P-value.
t-statistic value = 0.153 , alpha=0.05 then
p-value for two tail test = 0.879 which is greater than n = 0.05(0.879 > 0.05) So we do not reject H0.
4)Make a conclusion about the null hypothesis and a final conclusion that addresses the original claim.
we concluded that the null hypothesis not rejected, at 95 % confidence H0:u1= u2 accepted.
that is we accept the claim that women and men have the same mean diastolic blood pressure.at 0.05 significance level.
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Solve the equation 6(8x-2)=32
Write a decimal equivalent to the nearest hundredth. Put a zero before the decimal point.
2/9 = ____
Answer: 2/9 = 0.22
Step-by-step explanation: This is rounded to the nearest hundredth which is in the 2nd place after the decimal point.
Assuming that heights of professional male tennis players follow a bell-shaped distribution, arrange in ascending order.
I. A height with a z-score of 1
II. A height with a percentile rank of 80 percent
III. A height at the third quartile Q₃
(A) I,II,III
(B) I,III,II
(C) II,I,III,
(D) III,I,II
(E) III,II,I
A height with a z-score of 1 is the lowest, a height with a percentile rank of 80 percent is in the middle, and a height at the third quartile Q₃ is the highest.
A bell-shaped distribution is also known as a normal distribution. It is a symmetrical distribution where the mean, median, and mode are all equal. The distribution has two tails that extend out from the mean in either direction. A height with a z-score of 1 is the lowest because a z-score of 1 is one standard deviation below the mean. A height with a percentile rank of 80 percent is in the middle because it is the 80th percentile, meaning that 80 percent of the data is lower than it. A height at the third quartile Q₃ is the highest because it is the 75th percentile, meaning that 75 percent of the data is lower than it. In ascending order, the heights arranged are III, II, and I.
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9. MGSE7.EE.3: Game Stop is having a sale on last year's top games. For each game purchase,
the second game is 75% off. The original price of each game is $30.50. How much would 4
games cost?
A. $76.25
B. $91.50
C. $106.75
$122.00
$76.25 will be the total cost of four games including discount.
What is percent?Percent, A relative value that specifies one hundredth of an arbitrary amount. A percentage (representing 1%) is 1 in 100. So 100% represents the whole and 200% represents double the amount.
For example, 1 percent of 1,000 chickens is 1/100 of 1,000 or 10 chickens. 20% of the crowd is 20/100 1,000 or 200.
For the given case:
Original cost of each game = $30.50
75% of $30.50
(75/100) × 30.50
= $22.875
The cost of second game = 30.50 - 22.875
= $7.625
Since, 4 games are bought at once; this means 2 set of games with one on original price and another on discount price each.
Total cost of four games:
2 games at original price = 2 × 30.5
= $61
2 games at discount price = 2 × 7.625
= $15.25
Total = $61 + $15.25
= $76.25
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Determine whether the relation is a function. Explain.
(1,3), (2,4), (3,5), (1,5), (4,6)
Answer:
yes it is function.........
Find the solution of the inequality 5 > r – 3.
Answer:
8<r
Step-by-step explanation:
5 > r – 3
+3 +3
add 3 to both sides
8>r
move variable to the left and flip the sign
8<r
give an example to show that the conclusion of the mean value theorem can fail if we drop the requirement that f be differentiable at every point in (a, b). give an example to show that the conclusion can fail if we drop the requirement of continuity at the endpoints of the interval.
No such c exist from 1 and hence mean value theorem fails, an example is given below.
Consider f(x)=[x] on [-1,1]
Then, f is not differentiable on x=0
Note that,
f'(x)=1 for x>0 and f'(x)=-1 for x<0
The Mean Value Theorem states that if a feature f is non-stop at the closed interval [a,b] and differentiable at the open interval (a,b), then there exists a factor c withinside the interval (a,b) such that f'(c) is same to the feature's common price of extrade over [a,b].
According to near value theorem we need,
a c (-1,1).
f'(c)= f(1)-f(-1)/1-(-1)
=|1|-|-1|/2
=1-1/2=0
i.e., f'(c)=0
While, no such c exist from 1 and hence mean value theorem fails.
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a sequential circuit with two d flip-flops a and b, two inputs, x and y, and one output z is specified by the following next-state and output equations:
DB will appear at outputs A, and B.
Step 1
D flip flop stands for a delay flip flop that produces output that is identical to input after a brief delay. The flip flop's input and next state are equal.
Z is an output, and x and x inputs.
x'y + xA = DA= A*
B* = x'B + xA
z=A (output is independent of input) (output is independent of input)
The next states are A(t+1) and B(t+1). We must provide the next state value to D of the D flip flop in order to obtain the next states A(t+1), and B(t+1) for A, B(present state). So the sequential circuit will be:
Step 2
To get the next state as A(t+1)= x’A + xA, B(t+1)= x’B + xA, we have to give them to DA, DB of D flip flop because for D flip flop whatever is given to DA, DB will appear at output A, B.
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The previous balance on Hazel Taylor's savings account is $876.21. She has
$2.39 in interest, $616.46 in deposits, and $298.06 in withdrawals. What is her
new balance?
Hazel's new balance in the account is $1197
How to determine her new balance?From the question, we have the following parameters that can be used in our computation:
Previous balance = $876.21.
$2.39 in interest,
$616.46 in deposits,
$298.06 in withdrawals
Her new balance can be calculated using
New balance = Previous balance + interest + deposits - withdrawals
Substitute the known values in the above equation, so, we have the following representation
New balance = 876.21 + 2.39 + 616.46 - 298.06
Evaluate
New balance = 1197
Hence, the new balance is $1197
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a small island is 3 miles from the nearest point p on the straight shoreline of a large lake. if a woman on the island can row a boat 3 miles per hour and can walk 4 miles per hour, where should the boat be landed in order to arrive at a town 7 miles down the shore from p in the least time? let x be the distance (in miles) between point p and where the boat lands on the lakeshore.
x be the distance (in miles) between point p and where the boat lands on the lakeshore.x = (7 - 3)/(3/4) = 4.5 miles.The boat should be landed 4.5 miles down the shore from point p.
The woman needs to get to a town 7 miles down the shore from point P. She can row the boat 3 miles per hour, and walk 4 miles per hour. The boat needs to be landed 4.5 miles down the shore from point P so she can get to the town in the least time. To figure out this distance, we subtract the 3 miles between the island and point P from the 7 miles between the town and point P. This gives us 4 miles. We then divide 4 miles by the ratio of her rowing rate (3) to her walking rate (4) to get 4.5 miles. Therefore, the boat should be landed 4.5 miles down the shore from point P.
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KADS, Incorporated has spent $490,000 on research to develop a new computer game. The firm is planning to spend $290,000 on a machine to produce the new game. Shipping and installation costs of the machine will be capitalized and depreciated using bonus depreciated; they total $59,000. The machine has an expected life of three years, a $84,000 estimated resale value, and falls under the MACRS seven-year class life. Revenue from the new game is expected to be $690,000 per year, with costs of $340,000 per year. The firm has a tax rate of 21 percent, has an opportunity cost of capital of 12 percent, and expects net working capital to increase by $145,000 at the beginning of the project. What will the cash flows for this project be? Note: Negative amounts should be indicated by a minus sign. Round your answers to 2 decimal places.
Year 0 1 2 3
FCF ___ ___ ___ ___
Answer:
To find the cash flows for this project, we need to calculate the net income for each year and then subtract any investments in net working capital and capital expenditures.
In year 0, the firm will incur the following expenses:
- Research and development costs: $490,000
- Cost of the machine: $290,000
- Shipping and installation costs: $59,000
- Increase in net working capital: $145,000
Total expenses in year 0: $490,000 + $290,000 + $59,000 + $145,000 = $984,000
The net income in year 0 is equal to the revenue minus the costs: $690,000 - $340,000 = $350,000.
Therefore, the cash flow in year 0 is:
FCF0 = $350,000 - $984,000 = -$634,000
In years 1, 2, and 3, the firm will incur the following expenses:
- Cost of goods sold: $340,000
- Depreciation: (This can be calculated using the MACRS depreciation schedule. For the first year, the depreciation will be 14.29% of the cost of the machine, or 0.1429 * $290,000 = $41,461. The depreciation in subsequent years can be calculated using the same method.)
- Increase in net working capital: $145,000
Total expenses in each of years 1, 2, and 3: $340,000 + Depreciation + $145,000
The net income in each of these years is equal to the revenue minus the costs: $690,000 - (Total expenses)
The cash flow in each of these years is equal to the net income minus any investments in net working capital and capital expenditures: Net income - Increase in net working capital - Depreciation
Substituting the appropriate values, we get:
Year 1: FCF1 = $690,000 - ($340,000 + $41,461 + $145,000) = $163,539
Year 2: FCF2 = $690,000 - ($340,000 + Depreciation + $145,000)
Year 3: FCF3 = $690,000 - ($340,000 + Depreciation + $145,000)
Note that the depreciation in years 2 and 3 can be calculated using the same method as in year 1.
So, the cash flows for this project are:
Year 0 1 2 3
FCF -$634,000 $163,539
I hope this helps!
Which points define the solution set of this linear-quadratic system of equations?
Answer:
C. C and E
Step-by-step explanation:
As per the graph, both points C and E are points that are located on both the red and purple graphs.
Answer:
C. point C and point E
Step-by-step explanation:
The solutions to the given linear-quadratic system of equations are the points where the line and curve intersect.
From inspection of the given graph, the points of intersection are:
point C and point EThe mean s recorded by 27 candidates happened to be 64%. During inspection it was found that one candidate actually scored 74% s instead of 47% recorded in the result sheet. What will be the correct mean?
Step-by-step explanation:
the mean score is the sum of all scores divided by the number of scores.
64 = (s1+s2+s3+...+s26+s27)/27
64×27 = (s1+s2+s3+...+s26+s27)
1728 = (s1+s2+s3+...+s26+s27)
now, one of the scores changes from 47 to 74. that means the sum of all scores is increased by 74-47 = 27.
so, we get
1728 + 27 = (s1+s2+s3+...+s26+s27+27)
1755 = (s1+s2+s3+...+s26+s27+ 27)
we still have 27 scores, so for the updated mean we have to divide both sides by 27 again :
1755/27 = (s1+s2+s3+...+s26+s27+27)/27
65 = (s1+s2+s3+...+s26+s27+27)/27
the correct mean is therefore 65%.
Evaluate the given integral by changing to polar coordinates. yex dA, where R is the region in the first quadrant enclosed by the circle x2 y2 = 64
After evaluating the given integral by changing to polar coordinates we get [tex]7 e^8-31[/tex]
As per the details shared in the above question it is given that,
[tex]$x^2+y^2=64$[/tex]
[tex](x, y)=r(\cos \theta, \sin \theta)[/tex]
[tex]d A=r d r d \theta.[/tex]
Further solving above we get,
[tex]r^2=64 \\\Rightarrow & r=8 . \\0 \leq r \leq 8 ; & 0 \leq \theta \leq \pi / 2[/tex]
The polar coordinate,
[tex]\iint_{R} y e^x d A=\int_0^{\frac{\pi}{2}} \int_0^8 r^2 \sin \theta e^{r \cos \theta} d r d \theta[/tex]
[tex]& =\int_0^8 \int_0^{\frac{\pi}{2}} r^2 \sin \theta e^{r \cos \theta} d \theta ; \\& =\int_0^8\left[-r e^{r \cos \theta}\right]_0^{\pi / 2} d r[/tex]
[tex]=\int_0^8 r\left[-e^{r \cos \left(\frac{\pi}{2}\right)}+e^\alpha\right] d r .[/tex]
[tex]=\int_0^8 r e^r d r-\left.\frac{r^2}{2}\right|_0 ^8[/tex]
Now substituting the value we get,
[tex]7 e^8-31[/tex]
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Note the correct question should be ,
Evaluate the given integral by changing to polar coordinates.
[tex]\iint_R y e^x d A[/tex] , where R is the region in the first quadrant enclosed by the circle [tex]x^2+y^2=64[/tex].
A consumer products company is formulating a new shampoo and is interested
in foam height (in millimeters). Foam height is approximately normally distributed and has a
standard deviation of 20 millimeters. The company wishes to test H0 : µ = 175 millimeters vs
H1 : µ > 175 millimeters, using the results of n = 10 samples.
a) Find the type I error probability α if the critical region is ¯x > 185.
b) Find the boundary point of the acceptance region if α = 0.05
Probability can be used to make predictions or decisions in a variety of situations, such as in gambling, finance, and science. In these situations, probabilities can be calculated based on statistical data or by using mathematical models.
a) To find the type I error probability, we need to find the probability of rejecting the null hypothesis (H0) when it is actually true. In this case,
Since the foam height is approximately normally distributed with a standard deviation of 20 millimeters and we are taking a sample of size n = 10, the standard error of the mean is:
20 : [tex]\sqrt{10}[/tex] = 6.3 millimeters.
This means that the mean of the sampling distribution of the mean is approximately normally distributed with a mean of 175 millimeters and a standard deviation of 6.3 millimeters.
To find the probability of rejecting the null hypothesis when it is actually true, we can use a z-score.
In this case, the z-score for a value of 185 millimeters is:
(185-175)/6.3 = 1.6
We can use a z-table or a calculator to find the probability of getting a value greater than 1.6 standard deviations above the mean in a normal distribution. This probability is equal to 1 - the probability of getting a value less than or equal to 1.6 standard deviations above the mean, which is:
1 - 0.9495 = 0.0505.
Therefore, the type I error probability, or the probability of rejecting the null hypothesis when it is actually true, is 0.0505 if the critical region is x > 185.
b) To find the boundary point of the acceptance region if alpha = 0.05, we need to find the critical value that corresponds to a type I error probability of 0.05.
To find the critical value, we can use the z-score formula:
z = (x - mean) / standard deviation
We can rearrange this formula to solve for x:
x = z * standard deviation + mean
Since we want the probability of getting a sample mean greater than x to be 0.05, we can use a z-table or a calculator to find the z-score that corresponds to a probability of 0.05 in a normal distribution. This z-score is approximately 1.645.
Substituting this value into the formula above, we get:
x = 1.645 * 6.3 + 175 = 185.095
Therefore, the boundary point of the acceptance region if alpha = 0.05 is approximately 185.095 millimeters.
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Help!!!
A spinner is divided into five equal sections labeled 1, 2, 3, 4, and 5. Another spinner
is divided into two equal sections labeled A and B. Nigel will spin each spinner one
time.
How many of the possible outcomes have a number less than 3 or an A?
Hint: first write out the sample space (you should have 10)
Which samples have a number less than 3 or an A?
Answer: 7
Step-by-step explanation: cause I took the quiz…
If the average of 5 positive integers is 60 and the difference between the largest and the smallest of these 5 numbers is 10, what is the maximum value possible for the largest of these 5 integers? *
The largest value of these integers is 68
We are given that the average of 5 positive integers is 60
Let the five numbers be [tex]x_{1}, x_{2}, x_{3}, x_{4}, x_{5}[/tex]
Therefore,
[tex]\frac{x_{1}+ x_{2}+ x_{3}+ x_{4}+ x_{5}}{5} = 60[/tex]
[tex]{x_{1}+ x_{2}+ x_{3}+ x_{4}+ x_{5}} = (60)(5)[/tex]
[tex]{x_{1}+ x_{2}+ x_{3}+ x_{4}+ x_{5}} = 300[/tex]
Now, let the first (smallest) number be [tex]x[/tex]
Therefore, the last (largest) number will be equal to [tex]x + 10[/tex]
To maximise a number in the sum equation, we'll have to minimise all the other numbers.
Therefore, let it be
[tex]x + x + x + x + (x + 10) = 300[/tex]
[tex]5x + 10 = 300[/tex]
[tex]5x = 290[/tex]
Therefore, [tex]x = 58[/tex]
Now, we took the largest number as [tex](x + 10)[/tex]
Therefore, the largest number is (58 + 10)
The correct answer is 68
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twice the sum of a number and 3 is equal to three times the diffrence of the number and 3 find the number.
When "twice the sum of a number and 3 equals three times the difference of the number and 3," the value of expression is 6.
What is expression?In mathematics, an expression is a phrase that has at least two numbers or variables and at least one arithmetic operation. Addition, subtraction, multiplication, or division are all examples of math operations. A number, a variable, or a combination of numbers, variables, and operation symbols constitutes an expression. An equation consists of two expressions joined by an equal sign.
Here,
let x be the number,
2x+3=3x-3
x=6
The value of expression is 6 when "twice the sum of a number and 3 is equal to three times the difference of the number and 3."
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Alfonso bought 4 movie tickets for $29. Write an equation relating the total cost, y, to the number of tickets bought, x. Write the equation in the form ykx.
Graph the following points on a coordinate plane:
A (3,1)
B (3,-2)
C (-2,-2)
D (-2, 1)
Area of a rectangle = Length x Width
The area of this rectangle is
sq units
Answer:
Area = Length * Width
Area = 5 units * 3 units
Area = 15 units²
(see attachment picture for the graph)
Step-by-step explanation:
heart and star pls <3 brainliest will be appreciated <3(っ◔◡◔)っ -{ elyna s }-Answer: 15 sq units
Step-by-step explanation:
1. graph it out, this will help visually.
2. find the distance/difference between the y's of (3,1) and (3, -2) to find the width, or (-2, -2) and (-2, 1). it is 3.
3. find the distance between the x's of (3,1) and (-2,1), or (3, -2) and (-2, -2). the distance is 5.
4. multiply the lengths together to find the area. :)
Suppose 16 coins are tossed. Find the probability of getting the following result using the binomial probability formula and the normal curve approximation.
Exactly 7 heads
Binomial probability =
(Round to 4 decimal places).
On solving the provided question we can say that he probability of getting the following result using the binomial probability formula and the normal curve approximation is P(4, 16, 1/2) = 0.0280.
What is Probability?Probability is the possibility or chance that something will happen. Probability = how many different paths there are to success/the total number of events that might occur. A coin flip, for instance, has a 50% chance of coming up heads since there is only one method to acquire a head and there are a total of two possible outcomes (a head or tail). P(heads) = 1/2 is the formula.
P(4,16,1/2) = C(16,4)(1/2) 4(1 - 1/2)16 - 4 = C(16,4)(1/2)16 = 0.02777;
by help of table - P(4, 16, 1/2) = 0.0280
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the augmented matrix is given for a system of equations. if the system is consistent, find the general solution. otherwise state that there is no solution.
The given matrix has no solution.
What is the system of equations?
system of equations, or simultaneous equations, In algebra, two or more equations to be solved together (i.e., the solution must satisfy all the equations in the system). For a system to have a unique solution, the number of equations must equal the number of unknowns. Even then a solution is not guaranteed.
According to the question, the matrix is:
[tex]\left[\begin{array}{ccc}1&-5&-4\\0&0&7\\ \end{array}\right][/tex]
The linear equation will be:
x - 5y = -4
and
0x + 0y = 7
i.e,
0 = 7 (Not possible)
Hence, the above given matrix has no solution.
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suppose that a volcano is erupting and readings of the rate r(t) at which solid materials are spewed into the atmosphere are given in the table. the time t is measured in seconds and the units for r(t) are measured in tonnes (metric tons) per second. note that r(t) is increasing over the interval [0, 6]. t 0 1 2 3 4 5 6 r(t) 2 10 18 32 44 54 60 (a) give upper and lower estimates (in tons) for the total quantity q(6) of erupted materials after six seconds. (use six equal subintervals.)
The upper and the lower estimates (in tons) for the total quantity q(6) of erupted materials after six seconds is 224 and 166 metric tons respectively.
The volcano is erupting and readings of the rate r(t) at which solid materials are spewed into the atmosphere are given in the table
Assuming that r(t) is increasing for all t in [0,6], a lower estimate is obtained by using left endpoints and an upper estimate is achieved by using right endpoints.
Lower estimate: r(0)(1)+r(1)(1)+r(2)(1)+r(3)(1)+r(4)(1)+r(5)(1)
= 2+12+20+36+44+52 = 166 metric tons
Upper estimate: r(1)(1)+r(2)(1)+r(3)(1)+r(4)(1)+r(5)(1)+r(6)(1)
= 12+20+36+44+52+60 = 224 metric tons
Midpoint estimate: r(1)(2)+r(3)(2)+r(5)(2)
= 24+72+104 = 200 metric tons
Therefore, the upper and the lower estimates (in tons) for the total quantity q(6) of erupted materials after six seconds is 224 and 166 metric tons respectively.
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what percent of 70 is 63
Answer:
90%
Step-by-step explanation:
63 of 70 can be written as:
63/70
To find percentage, we need to find an equivalent fraction with denominator 100. Multiply both numerator & denominator by 100
63/70 × 100/100 = (63 × 100/70) × 1/100 = 90/100
Therefore, the answer is 90%
If you are using a calculator, simply enter 63÷70×100 which will give you 90 as the answer.
large sample significance tests for a single proportion are based on the .group of answer choices a. statistic binomial b. distribution test
c. uniform distribution
Large sample significance tests for a single proportion are based on the statistic binomial. the correct option is A
What is significance tests?Significance test can be defined as tests for statistical significance indicate whether observed differences between assessment results occur because of sampling error. Significance tests give a formal process for using sample data to evaluate the likelihood of some claim about a population value.
Therefore a Large sample significance tests for a single proportion are based on the statistic binomial.
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Sketch the graph of each function.
1)f(x)= -2 x-1, & ,x≤2
-x+4,x>2
The graph of each given piecewise function is shown below .
In the question,
a piece wise function is given ,
we have to sketch the graph of the piecewise function ,
the piecewise function f(x) is = [tex]\left \{ {-2x-1 , {x\leq 2} \atop -x+4 , {x > 2}} \right.[/tex]
when x ≤ 2 ,
we put x =2 , we get , f(2) = -2*2-1 = -5 .
So , the point is (2,-5) .
when we put x = 0 , f(0) = -2*0 - 1 = -1
so , the second point is (0,-1) .
Now , for x>2
we put x= 4 , we get f(4) = 0
the point will, be (4,0)
we put x = 6 , we get , f(6) = -2 .
the point will be = (6,-2) .
On plotting all the four points , we get the graph as sketched below .
Therefore , the given piecewise function's graph is shown below.
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A baseball team plays in a stadium that holds 72000 spectator. With the ticket price at dollars8 average attendance has been 30000. When the price dropped to dollars5, the average attendance rose to 36000. Find the demand function P(x), where x is the number of the spectators. (Assume p(x) is linear.) p(x) = How should ticket prices be to maximize revenue? Price =
To generate the most money, the stadiums might set the price at $11.50.A mathematical equation known as a demand function expresses consumer demand as a function of a product or service's pricing and .
How does demand function work?The price that customers are prepared to pay and the amount that they're willing to purchase are generally correlated through a demand function. Since the profit function is the sum of these two amounts, this relationship can be converted into one.
m=5−836−30=−36=−1
This is the number of tickets sold that will maximize the revenue because the second derivative is, in fact, negative. Let's go back to the demand function to determine the appropriate ticket price.
R=−x+23\s−x+23=0x=23
R=−1\sp=−12(23)+23=11.5
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An offshore oil well is 3 kilometers north off the coast that runs directly east to
west. A point, E, is labeled on the coast directly south of the oil well. The refinery
is located 4 kilometers east of point E. Laying pipe in the water costs $500,000 per kilometer and laying pipe on the coast costs $300,000 per kilometer.
What path should the pipe follow in order to minimize the cost?
Show all process leading to your answer.
Answer:
come ashore 1.75 km west of the refinerycost: $2.4 millionStep-by-step explanation:
You want to know the least-cost path of a pipeline from 3 km offshore to 4 km east of the closest point on shore. The cost of pipe in the water is $0.5M/km, and onshore is $0.3M/km.
Generic solutionThe solution in the general case for this kind of question is that the sine of the angle between the route directly to shore and the least-cost route is the ratio of the onshore cost to the in-water cost.
sin(α) = (0.3M/km)/(0.5M/km) = 3/5
The distance from the closest point onshore (E) to the least-cost onshore point is the product of the distance from shore and the tangent of this angle:
distance downshore = (3 km)·tan(arccos(3/5)) = 2.25 km
Since the refinery is 4 km downshore, this point is 1.75 km from the refinery.
The pipe should follow a direct path underwater from the well to a point 1.75 km west of the refinery, then downshore the remaining 1.75 km to the refinery.
Cost equationIf x represents the distance from the refinery to the point where the pipeline comes ashore, the cost of the route in millions will be ...
c = 0.5√(3² +(4-x)²) +0.3x
where the square root function is calculating the underwater length of the pipeline using the Pythagorean theorem.
MinimizationThis cost function will be minimized when its derivative with respect to x is zero.
c' = 0.5(1/2)(2x -8)/√(x² -8x +25) +0.3 = 0
0 = 0.5(x -4) +0.3√(x² -8x +25) . . . . . . numerator of the combined terms
4 -x = 0.6√(x² -8x +25) . . . . multiply by 2, add 4-x
16 -8x +x² = 0.36x² -2.88x +9 . . . . . . square both sides
0.64x² -5.12x +7 = 0 . . . . . subtract the right side
x = (5.12 -√(5.12² -4(0.64)(7)))/(2(0.64)) = 2.24/1.28 = 1.75
The route with lowest cost comes ashore 1.75 km from the refinery.
__
Additional comment
The cost of the pipeline will be ...
c = 0.5√(9 +(4-1.75)²) +0.3(1.75) = 0.5√14.0625 +0.525
c = 2.4 . . . . million dollars
The attached graph shows the cost function has a minimum of 2.4M at a distance of 1.75 km from the refinery. The cost is only 2.5M if the pipe runs directly from the well to the refinery (x=0).
What is the scale factor of the geometric sequence?
200, 100, 50, 25, . . .
Enter your answer as a decimal in the box.
The scale factor of the geometric sequence 200, 100, 50, 25 is ½
What is scale factor?A scale factor is defined as the ratio between the scale of a given original object and a new object, which is its representation but of a different size.
Therefore scale factor= new dimension/ original dimension.
In the sequence 200, 100, 50, 25...., as we take 200 as the original dimension and 100 as the new dimension. From these we can discover that the object is decreasing in shape
Therefore the scale factor = new dimension/ original dimension
scale factor = 100/200
scale factor = ½
therefore the scale factor of the sequence is ½
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