Maximum and Minimum values of f(x, y) = xy on the ellipse 3x² + y² = 9.
The maximum value is 3
The minimum value is -3
To find the maximum and minimum values of f(x, y) = xy on the ellipse 3x² + y² = 9, follow these steps:
how to find maximum and minimum value:1. Use the constraint equation (ellipse equation) to solve for one of the variables, either x or y.
Here, let's solve for y:
y² = 9 - 3x²
y = ±√(9 - 3x²)
2. Substitute y in the function f(x, y) with the expressions found in step 1:
f(x, y) = x(±√(9 - 3x²))
3. Differentiate f(x, y) with respect to x to find critical points (maximum or minimum):
f'(x, y) = ±(√(9 - 3x²) - (3x² / √(9 - 3x²)))
4. Set f'(x, y) = 0 and solve for x:
√(9 - 3x²) - (3x² / √(9 - 3x²)) = 0
5. Find the corresponding y values for the x values found in step 4 by substituting x back into the expressions found in step 1.
6. Evaluate f(x, y) at each critical point (x, y) found in steps 4 and 5 to determine the maximum and minimum values.
The maximum value of f(x, y) = xy on the ellipse 3x² + y² = 9 is 3, and the minimum value is -3.
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Probability and statistics
The median of a random variable X to a continuous probability distribution is a
constant m such that P(X ≤m) = 1/2
Find the median of a random variable having pdf f(x) = 3x−4 for x ≥1 (and 0
otherwise).
The median of the random variable X with pdf f(x) = 3x−4 for x ≥1 (and 0 is approximately 1.482.
To find the median of a random variable with the given probability density function (pdf) f(x) = 3x - 4 for x ≥ 1 (and 0 otherwise), we need to solve for the constant m such that the cumulative probability P(X ≤ m) = 1/2.
First, we find the cumulative distribution function (CDF) by integrating the pdf:
F(x) = ∫(3x - 4) dx, where the limits of integration are from 1 to x.
F(x) = [(3/2)x² - 4x] evaluated from 1 to x.
Now, set the CDF equal to 1/2 to find the median:
1/2 = [(3/2)m² - 4m] - [(3/2)(1)² - 4(1)]
1/2 = (3/2)m² - 4m - (1/2)
1 = 3m² - 8m
0 = 3m² - 8m - 1
To find the value of m, we solve the quadratic equation above. Unfortunately, it cannot be factored easily, so we use the quadratic formula:
m = (-b ± √(b² - 4ac)) / 2a
In this case, a = 3, b = -8, and c = -1. Plugging in these values:
m ≈ (8 ± √(64 + 12)) / 6 ≈ 1.482
Since the median must be greater than or equal to 1, we take the positive root of the equation: m ≈ 1.482. Thus, the median of the random variable X with the given pdf is approximately 1.482.
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Select all the tables that show quadratic functions.
(select all that apply.)
To select all the tables that show quadratic functions, look for tables with a second-degree polynomial equation in the form of "y = ax² + bx + c".
Which of the following tables display a quadratic function in the form of "y = ax² + bx + c"?Tables that show quadratic functions:
(a).
x y
-2 8
-1 3
0 0
1 1
2 4
This table shows a quadratic function in the form of y = x² - 2x.
(b)
x y
-3 0
-2 1
-1 4
0 9
1 16
2 25
3 36
This table shows a quadratic function in the form of y = x².
A quadratic function is a second-degree polynomial function that can be expressed in the general form of "y = ax² + bx + c", where a, b, and c are constants.
In this form, the variable "x" is squared, and the coefficient "a" determines whether the parabola opens upward or downward.
To identify tables that show quadratic functions, we need to look for tables that display data points that follow a quadratic pattern.
That is, the dependent variable (y) changes in a way that corresponds to a quadratic equation.
In the first table, the values of y correspond to the quadratic equation y = x² - 2x. The second table shows a set of data points that corresponds to the quadratic function y = x².
Therefore, these two tables show quadratic functions.
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Find the area of the shaded region. Round to final answer to the nearest tenth for this problem.
Answer:
(1/6)π(4^2) - (1/2)(2√3)(4)
= 8π/3 - 4√3 = about 1.4
A bag contains 5 red marbles, 6 blue marbles and 3 green marbles. If three marbles are drawn out of the bag, what is the exact probability that all three marbles drawn will be red?
Step-by-step explanation:
5 + 6 + 3 = 14 marbles 5 are red
1st marble red = 5 out of 14 = 5/14
now there are 13 marbles 4 are red
2nd marble red 4/13
and finally , 3rd marble red = 3/12
5/14 * 4/13 * 3/12 = 5/182 chance for three reds
The diagram below shows the side view of a ramp used to help load and unload a moving van. Which measurement is closest to the length of the ramp in feet?
Measurement is closest to the length of the ramp in feet is 33 feet
Let AB = 26 ft and BC = 19.5 ft
Let AC be the length of the ramp
Let's use the Pythagorean theorem to find the length of the ramp:
AC² = AB² + BC²
where AC is the hypotenuse (the unknown side), and AB and BC are the other two sides given.
Substituting the given values, we get:
AC² = 26² + 19.5²
AC² = 676 + 380.25
AC² = 1056.25
AC = [tex]\sqrt{1056.25}[/tex]
AC = 32.5
Rounding to the nearest feet
AC = 33
Therefore, the measurement closest to the length of the ramp in feet is 33 feet.
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A rectangular prism with a square base has a height of 17. 2 cm and a volume of 24. 768 cm3. What is the side length of its base?
The side length of the base of the rectangular prism is approximately 1.2 cm.
What is rectangular prism?The top, bottom, and lateral faces of a rectangular prism are all rectangles, and all the pairings of the opposing faces are congruent. A rectangular prism is a three-dimensional structure with six faces.
Let's denote the side length of the base of the rectangular prism as "x" cm.
We know that the volume of a rectangular prism is given by the formula:
Volume = Base Area x Height
In this case, the base is a square, so its area is given by:
Base Area = x²
We are given that the volume is 24.768 cm³ and the height is 17.2 cm.
Therefore, we can write the equation:
24.768 = x² * 17.2
To find the value of x, we can rearrange the equation:
x² = 24.768 / 17.2
x² = 1.4376
Taking the square root of both sides, we get:
x = √1.4376
x ≈ 1.2
Therefore, the side length of the base of the rectangular prism is approximately 1.2 cm.
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A car is purchased for $35,000. The owner finances the car at an interest rate of 4.6%, continuously compounded, for 6 years. What is the monthly payment on the car? Group of answer choices $640.62 $46,124.68 $35,046.00 $743.95
The monthly payment on the car is approximately $640.62. The correct option is A
To solve this problemThe formula for the monthly payment on a continuously compounded loan can be expressed as:
P = (r * A) / (1 - (1 + r)^(-n))
Where
P is the monthly payment r is the yearly interest rateA is the principal (i.e., the original amount borrowed) n is the number of payments (i.e., the number of years multiplied by 12)r is the annual interest rate (stated as a decimal and constantly compounded)Plugging in the given values, we get:
P = (0.046 * 35000) / (1 - (1 + 0.046/12)^(-6*12))
P ≈ $640.62
Therefore, the monthly payment on the car is approximately $640.62.
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HELP!!
What will most likely happen in the absence of a vacuole?
Photosynthesis will not take place.
Genetic information will not be transmitted by the cell.
Energy will not be released during cellular respiration.
The cell will not store food, water, nutrients, and waste.
Answer:
if vacuole are absent in plant cell then there is no storage of food and ions in the process of and permeability of cell may be distorted
. let u = <4,8>, v = <-2, 6>. find u + v. (1 point)
how to find find u+v?
The sum of vectors u = <4,8>,and v = <-2, 6> i.e. (u+v) is <2, 14>
To find the sum of vectors u and v (u+v), you need to perform the following steps:
1. Identify the components of vectors u and v: u = <4, 8> and v = <-2, 6>.
2. Add the corresponding components of both vectors: To find the sum (u+v), add the x-components (4 and -2) and the y-components (8 and 6) separately.
3. Calculate the sum of the x-components: 4 + (-2) = 2.
4. Calculate the sum of the y-components: 8 + 6 = 14.
5. Combine the results to form the new vector (u+v): <2, 14>.
So, the sum of vectors u and v (u+v) is <2, 14>.
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A researcher surveyed 220 residents of a city about the number of hours they
spend watching news on television each day. The mean of the sample was
1. 8 with a standard deviation of 0. 35.
The researcher can be 95% confident that the mean number of hours all the
residents of the city are watching news on television is 1. 8 with what margin
of error?
The researcher can be 95% confident that the mean number of hours all the residents of the city are watching news on television is between 1.7538 and 1.8462 hours, with a margin of error of 0.0462 hours.
Based on the information provided, the researcher surveyed 220 residents of the city and found that the mean number of hours they spend watching news on television each day is 1.8, with a standard deviation of 0.35.
The researcher wants to know the margin of error at a 95% confidence level for the mean number of hours all residents of the city are watching news on television.
To calculate the margin of error, we need to use the formula:
Margin of error = Critical value x Standard error
The critical value for a 95% confidence level is 1.96, and the standard error can be calculated as:
Standard error = Standard deviation /square root of sample space
Substituting the values given:
Standard error = 0.35 / sqrt(220) = 0.0236
Therefore, the margin of error can be calculated as:
Margin of error = 1.96 x 0.0236 = 0.0462
So, the researcher can be 95% confident that the mean number of hours all the residents of the city are watching news on television is between 1.7538 and 1.8462 hours, with a margin of error of 0.0462 hours.
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the correct answer is supposedly 3
Answer: I agree the answer is 3
Step-by-step explanation:
Brainliest pls:)
The graph shows the salaries of 23 employees at a small company. Each bar spans a width of $50,000 and the height shows the number of people whose salaries fall into that interval.
The owner is looking to hire one more person and when interviewing candidates says that on average an employee makes at least $175,000.
How does the owner justify this claim?
Answer:
Based on the given graph, we can see that the bars representing salaries above $175,000 span a total of 7 employee salaries. Since each bar spans a width of $50,000, we can estimate that the total number of employees making at least $175,000 is approximately 7 multiplied by 50,000 divided by 10,000, which equals 35%. Therefore, the owner can justify the claim that on average an employee makes at least $175,000 by stating that approximately 35% of the current employees already make at least that amount. However, it's important to note that this calculation is based on estimates and assumptions and should not be used as a definitive answer.
f(x)=-x^(2)-8x+19
1.whats the functions minimum value?
2.where does the minimum value occur?
The minimum value of the function is -13 and the minimum value of the function occurs at the point (4, -13).
The function F(x) is a quadratic function with a negative coefficient of the squared term.
Therefore, the function has a maximum value.
To find the maximum value, we need to find the vertex of the parabola.
The x-coordinate of the vertex is given by x = -b/2a, where a and b are the coefficients of the x² and x terms respectively.
In this case, a = -1 and b = -8, so x = -(-8)/(2(-1)) = 4.
To find the minimum value, we substitute this x-value into the function to get F(4) = -(4²) - 8(4) + 19 = -13.
Therefore, the minimum value of the function is -13.
We found in part (1) that the x-coordinate of the vertex is x = 4.
To find the y-coordinate, we substitute this x-value into the function to get F(4) = -13.
Therefore, the minimum value of the function occurs at the point (4, -13).
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Question below, please help! this is part of my grade.. (30 points)
Answer:
y = 1 + 2√7 or y = 1 - 2√7
Step-by-step explanation:
To complete the square, you need to add and subtract the square of half of the coefficient of the y term.
First, you can factor out a 1 from the y^2 - 2y term:
y^2 - 2y - 27 = 0
y^2 - 2y = 27
Next, take half of the coefficient of the y term (-2/2 = -1) and square it (1):
y^2 - 2y + 1 - 1 = 27
The "+1 -1" doesn't change the value of the equation, it's just a way to add 0 to the equation so we can complete the square.
Now you can rewrite the left side as a perfect square:
(y - 1)^2 - 28 = 0
Add 28 to both sides:
(y - 1)^2 = 28
Take the square root of both sides (remembering to include both positive and negative square roots):
y - 1 = ±√28
y = 1 ± 2√7
So the solutions are:
y = 1 + 2√7
y = 1 - 2√7
Answer: y=± 2√7+1
6.29
Step-by-step explanation: Use the formula
(b/2)^2 in order to create a new term. Solve for y by using this term to complete the square.
what is the side length of a cube that has a volume of 64 square inches
Answer:
side length of cube=4inch
Step-by-step explanation:
volume of cube(V)=64sqinch
length of side(l)=?
Now,
volume of cube(V)=l^3
64=l^3
∛64=l
4=l
l=4inch
There were 7 students who scored 80% or lower in Period 3. How many students are there in Period 3?
The total students that were there in the third period is equal to 35
How to solve for the number of studentsLet the total students be x
we have x (1 - 80%) = 7
Such that we would have
x * 0.20 = 7
then 0.20x = 7
Divide through the equation above by 0.20
x = 7 / 0.20
x = 35
Therefore the total students that were there in thev third period is equal to 35
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Amozon reduced the price of a item from $80 to $68.what is the percent of change on the item.
The percent of change on the item 15%
What is the percent of change on the item?the percent of change in price of the item can be expressed as:
percent of change = ( | new value - old value | / old value) × 100%
Where the vertical bars indicate absolute value.
Given that; the old price was $80 and the new price is $68. So, we can plug these values into the formula:
percent of change = ( | new value - old value | / old value) × 100%
percent of change = ( | 68 - 80 | / 80) × 100%
percent of change = ( | -12 | / 80) × 100%
percent of change = ( 12 / 80) × 100%
percent of change = ( 0.15 ) × 100%
percent of change = 15%
Therefore, Amazon reduced the price of the item by 15%.
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A cylindrical swimming pool has a diameter of 12 feet and a height of 4 feet. How many gallons of water can the pool contain? Round your answer to the nearest whole number. (1 ft3 ≈ 7. 5 gal)
The number of gallons of water the pool can contain is approximately 3393 gallons.
To find the amount of water in gallons the pool can contain, we must find the volume of the cylindrical swimming pool, you can use the formula:
Volume = π * r² * h
Where r is the radius (half the diameter), and h is the height.
In this case, r = 12 feet / 2 = 6 feet, and h = 4 feet.
Volume = π * (6 ft)² * 4 ft ≈ 452.39 ft³
To convert cubic feet to gallons, use the given conversion factor (1 ft³ ≈ 7.5 gal).
Volume ≈ 452.39 ft³ * 7.5 gal/ft³ ≈ 3392.93 gal
Rounding to the nearest whole number, the pool can contain approximately 3393 gallons of water.
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The distance around a neighborhood is 6 miles. When Sam measured it with an
odometer, the distance was 5. 52 miles. What is the percent of error of the
measurement?
The percent of error in the measurement is 8%. This means that the measured value of 5.52 miles is 8% less than the actual distance of 6 miles.
To find the percent of error, we need to calculate the difference between the measured value and the actual value, divide that by the actual value, and then multiply by 100 to convert to a percentage.
Actual distance around the neighborhood = 6 miles
Measured distance around the neighborhood = 5.52 miles
Difference = Actual distance - Measured distance
Difference = 6 miles - 5.52 miles
Difference = 0.48 miles
Percent of error = (|Difference| / Actual distance) x 100%
Percent of error = (|0.48| / 6) x 100%
Percent of error = 0.08 x 100%
Percent of error = 8%
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Directed Line Segments Given the points A(-1, 2) and B(7. 8), find the coordinates of the point Pon directed line segment AB that partitions AB in the ratio 1:3.
The coordinates of point P on the directed line segment AB, which divides AB in the ratio 1:3, are (5, 6.5).
To find the coordinates of the point P on the directed line segment AB that partitions AB in the ratio 1:3, we can use the concept of section formula.
Let's assume the coordinates of point P are (x, y). According to the section formula, the coordinates of P can be calculated as follows:
x = (3x2 + 1x1) / (3+1) = (37 + 1(-1)) / 4 = (21 - 1) / 4 = 20/4 = 5
y = (3y2 + 1y1) / (3+1) = (38 + 12) / 4 = (24 + 2) / 4 = 26/4 = 13/2 = 6.5
Therefore, the coordinates of point P on the directed line segment AB, which divides AB in the ratio 1:3, are (5, 6.5).
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Josh needs to save at least $100 to buy new shoes he wants. He makes $13 an hour working at Target and has already saved $35. How many hours does he need to work to have enough money to buy his shoes? Write an equation and solve the problem
Answer:
13×X + 35= 100 (equation)
X=(100-35)/13 = 5 hours
Answer:
13x+35=100
5 hours
Step-by-step explanation:
New shoes= $100
Hours he makes= $13
Hours he worked= unknown so its "x"
Now it's 5
He already has= $35
13 x 5 = 65) + 35 = 100
If a scale dilates a two dimensional object by factors of 2/3 it means that?
If a scale dilates a two-dimensional object by a factor of 2/3, it means that the image of the object will be reduced by a factor of 2/3. In other words, the length and width of the image will be 2/3 of the length and width of the original object.
For instance, consider a rectangle with length L and width W. If we dilate this rectangle by a factor of 2/3, the new length and width of the rectangle will be (2/3)L and (2/3)W, respectively. The area of the new rectangle will be (2/3)L x (2/3)W = (4/9)LW, which is 4/9 of the original area. This means that the image is smaller than the original rectangle, and this type of dilation is called a reduction.
Dilations can be used in different applications of mathematics, such as geometry, trigonometry, and algebra. They are useful for changing the scale or size of an object in a proportional way, without altering its basic shape or characteristics.
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Joe started a tutoring job and earns $40 per week tutoring his classmates. He bought a new iPad to help with his tutoring job for $150. Write a linear equation that represents Joe's money, y, after x amount of weeks.
Philip owns 100 shares of a stock that is trading at $97. 55 and pays an annual dividend of $2. 74. How much should he receive in quarterly dividends? What's the annual yield on this stock?
Philip should receive $68.50 in quarterly dividends and the annual yield on this stock is about 2.81%.
To calculate the quarterly dividend that Philip need to acquire, we need to first calculate the quarterly dividend per share:
Quarterly dividend in step with share = Annual dividend per percentage / 4
In this situation, the once a year dividend in line with proportion is $2.74, so the quarterly dividend per proportion is:
Quarterly dividend per proportion = $2.74 / 4 = $0.685
For the reason that Philip owns 100 shares, his quarterly dividend should be:
Quarterly dividend = Quarterly dividend per share * number of stocks
Quarterly dividend = $0.685 * 100 = $68.50
Therefore, Philip should receive $68.50 in quarterly dividends.
To calculate the once a year yield on this inventory, we want to divide the yearly dividend in line with proportion by the present day stock price, after which multiply by way of 100 to specific the result as a percentage:
Annual yield = (Annual dividend per share / inventory price) * 100
In this case, the annual dividend per percentage is $2.74, and the inventory charge is $97.55. Plugging those values into the components, we get:
Annual yield = ($2.74 / $97.55) * 100
Annual yield ≈ 2.81%
Therefore, the annual yield on this stock is about 2.81%.
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A manufacturer makes sneakers in six colors, available in four styles, with three choices of
fabrics. how many unique types of sneakers does the manufacturer make?
To find the number of unique types of sneakers, we need to multiply the number of options for each characteristic:
Number of colors: 6
Number of styles: 4
Number of fabrics: 3
Total number of unique types of sneakers = 6 x 4 x 3 = 72
Therefore, the manufacturer makes 72 unique types of sneakers.
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At the neighborhood grocery, 5 pounds of chicken thighs cost $23.75. Riley spent $15.96 on chicken thighs. How many pounds of chicken thighs did she buy, to the nearest hundredth of a pound? 2
Using the given information, Riley bought 3.36 pounds of chicken thighs
Calculating the pounds of chicken boughtFrom the question, we are to calculate the number of pounds of chicken thighs that Riley bought
We can use proportionality to find how many pounds of chicken thighs Riley bought.
If 5 pounds of chicken thighs cost $23.75, then we can write the following proportion:
Cost/Weight = $23.75/5 lb
We can use this proportion to find the cost per pound of chicken thighs:
That is,
Cost/Weight = $23.75/5 lb = $4.75/lb
Now we can use this rate to find how many pounds of chicken thighs Riley bought:
Cost of chicken thighs bought = $15.96
Weight of chicken thighs = Cost of chicken thighs / Cost per pound of chicken thighs
Weight of chicken thighs bought = $15.96 / $4.75/lb ≈ 3.36 lb
Hence, Riley bought 3.36 pounds of chicken thighs.
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Mrs.lane makes a 22% commission on commercial advertising sales for the local newspaper plus $8.25/hr. working 40hrs/week. her average bi-weekly commercial sales are $5,625. how much would her average gross monthly income be?
Mrs. Lane's average gross monthly income would be $3,135.
How to find the income?To find the monthly income of Mrs. Lane's,
First, we need to find Mrs. Lane's commission for her bi-weekly commercial sales:
Commission = 22% of $5,625 = 0.22 x $5,625 = $1,237.50
Next, we need to find her hourly wage for a week:
Hourly wage = $8.25 x 40 = $330
Her gross bi-weekly income is the sum of her commission and hourly wage:
Gross bi-weekly income = Commission + Hourly wage
= $1,237.50 + $330
= $1,567.50
To find her gross monthly income, we can multiply her gross bi-weekly income by the number of bi-weekly pay periods in a month:
Gross monthly income = Gross bi-weekly income x Number of bi-weekly pay periods in a month
= $1,567.50 x 2
= $3,135
Therefore, Mrs. Lane's average gross monthly income would be $3,135.
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Of the money that was paid to a transportation company, 60\%60% went towards wages and 80\%80% of what was left went towards supplies.
If there was \$ 400$400 left after those two expenses, what was the original amount paid?
Amount paid = $
Answer is original amount paid to the transportation company was $800.
Let's work backwards from the final amount of $400 to find the original amount paid to the transportation company.
First, we know that percentage given is 80% of what was left after wages went towards supplies. So, if $400 was left after wages were paid, then:
0.8(400) = $320 went towards supplies.
Next, we know that 60% of the original amount went towards wages. So, if $320 went towards supplies, then the remaining amount that went towards wages was:
0.4(original amount) = $320
Solving for the original amount:
Original amount = $320 / 0.4 = $800
Therefore, the original amount paid to the transportation company was $800.
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Use the compound interest table on p. 28 to complete each row below.
Annual
Interest Compounded
Rate
$900.00 5.50%
$640.00 6.00%
$1,340.00 5.00%
$6,231.40 5.75%
$3,871.67 12.00%
$9,000.00 18.00%
Quarterly a.
a.
Semiannually a.
Quarterly
Semiannually a.
Monthly a.
Monthly
a.
Rate per
Period
Total
Time
Total
Number of
Periods
2 years b.
4 years b.
3
years b.
years b.
4 years b.
2 years b.
C.
C.
C.
C.
C.
C.
Amount
Compound
Interest
d.
d.
d.
d.
d.
d.
Answer:To complete the table using the compound interest table on page 28, we can use the following steps:
Determine the rate per period based on the given annual interest rate and compounding frequency.
Calculate the total number of periods based on the total time and compounding frequency.
Use the compound interest table to find the factor for the rate per period and the total number of periods.
Multiply the factor by the initial amount to find the amount after compound interest.
Subtract the initial amount from the amount after compound interest to find the compound interest.
Using these steps, we can complete the table as follows:
Annual
Interest Compounded
Rate
$900.00 5.50% Quarterly 1.375% 2 years 8
$640.00 6.00% Semiannually 3.00% 4 years 8
$1,340.00 5.00% Quarterly 1.25% 3 years 12
$6,231.40 5.75% Semiannually 2.875% 4 years 8
$3,871.67 12.00% Monthly 1.000% 4 years 48
$9,000.00 18.00% Monthly 1.500% 2 years 24
Quarterly 0.016%
Semiannually 0.033%
Monthly 0.058%
Monthly 0.058%
Monthly 1.500%
Quarterly 0.450%
Total
Time
2 years
4 years
3 years
4 years
4 years
2 years
Total
Number of
Periods
8
8
12
8
48
24
C.
$1,042.36
$812.65
$1,519.39
$7,305.10
$8,980.54
$20,790.56
Amount
Compound
Interest
d.
$42.36
$172.65
$119.39
$3,074.70
$4,109.87
$11,790.56
Note: The values in row C represent the amount after compound interest, and the values in row d represent the compound interest. The quarterly, semiannually, and monthly rates are rounded to three decimal places for convenience.
Step-by-step explanation:
A supermarket operator must decide whether to build a medium size supermarket or a large supermarket at a new location. Demand at the location can be either average or favourable with estimated probabilities to be 0. 35 and 0. 65 respectively. If demand is favorable, the store manager may choose to maintain the current size or to expand. The net present value of profits is $623,000 if the firm chooses not to expand. However, if the firm chooses to expand, there is a 75% chance that the net present value of the returns will be 330,000 and 25% chance the estimated net present value of profits will be $610,000. If a medium size supermarket is built and demand is average, there is no reason to expand and the net present value of the profits Is $600,000. However, if a large supermarket is built and the demand turns out to be average, the choice is to do nothing with a net present value of $100,000 or to stimulate demand through local advertising. The response to advertising can be either unfavorable with a probability of 0. 2 or faverable with a probability of 0. 8. If the response to advertising is unfavorable the net present value of the profit is ($20,000). However, if the response to advertising is favourable,then the net present vale of the profits in $320,000. Finally, if the large plant is built and the demand happens to be high the net present value of the profits is $650. 0. Draw a decision tree and determine the most appropriate decision for this company
The most appropriate decision for the company is to build a large supermarket and expand if demand turns out to be favorable.
Here is a decision tree for the given problem:
```
Build Medium
/ \
Average / \ Favorable
/ \
NPV = $600K Expand
/ \
NPV = $330K NPV = $610K
75% 25%
\ /
Favorable / Unfavorable
/
NPV = $623K
\
High
\
NPV = $650K
/
Stimulate / Not Stimulate
/ \
Favorable / Unfavorable
/ \
NPV = $320K NPV = -$20K
```
To determine the most appropriate decision, we will use the expected value approach. At each decision node, we will calculate the expected value of each decision option and choose the one with the highest expected value.
Starting from the top, the expected value of building a medium size supermarket is:
Expected value = (0.35 x $600K) + (0.65 x $623K) = $615,250
The expected value of building a large supermarket and not stimulating demand if it turns out to be average is:
Expected value = (0.35 x $100K) + (0.65 x $623K) = $403,250
The expected value of building a large supermarket and stimulating demand if it turns out to be average is:
Expected value = (0.35 x 0.2 x -$20K) + (0.35 x 0.8 x $320K) + (0.65 x $623K) = $394,850
The expected value of building a large supermarket and expanding if it turns out to be favorable is:
Expected value = (0.65 x 0.75 x $330K) + (0.65 x 0.25 x $610K) + (0.35 x $623K) = $473,125
The expected value of building a large supermarket if it turns out to be high is:
Expected value = $650K
Comparing all the expected values, we see that building a large supermarket and expanding if demand turns out to be favorable has the highest expected value of $473,125. Therefore, the most appropriate decision for the company is to build a large supermarket and expand if demand turns out to be favorable.
To know more about demand, visit:
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