Financial Algebra 1st Edition Chapter 2 Modeling a Business
Page 104 Problem 1 Answer
Given: p=42, E=50q+80,000 and q=80p+100,000
To Determine the expense E
Use the equation for q q=80p+100,000.
Substitute 42 for q q=80(42)+100,000.
Simplify q=103,360
Use the equation for E E=50q+80,000
Substitute q E=50(103,360)+80,000
Simplify E=5,248,000
Therefore the expense is 5,248,000.
Page 105 Problem 2 Answer
Given:
To show :P=R−E
Graphically, profit is the vertical distance between the revenue and expense functions.
In Figure, the top of the curve(dot) hits the revenue graph at 40,380,000 when the price is about 25.
The bottom of the vertical line segment hits the expense graph at150,500
at the same price. The vertical length of this segment is 40,380,000−150,500=3887,500 and is the profit the company makes when price is about 25.
P=R−E where P is profit, R is revenue, and E is expenses
The greatest difference between revenue and expense denotes maximum profit.
Hence shown P=R−E
Page 106 Problem 3 Answer
A company produces a security device known as Toejack.
Toejack is a computer chip that parents attach between the toes of a child, so parents can track the child’s location at any time using an online system.
The company has entered into an agreement with an Internet service provider, so the price of the chip will be low.
To find as the price increases, what is expected to happen to the quantity demanded
It is pretty simple, as the price increases, the demand goes down.
As the price increases, the demand goes down.
Page 106 Problem 4 Answer
A company produces a security device known as Toejack.
Toejack is a computer chip that parents attach between the toes of a child, so parents can track the child’s location at any time using an online system.
The company has entered into an agreement with an Internet service provider, so the price of the chip will be low.
The horizontal axis represents price, and the vertical axis represents quantity
To find if the slope of the demand function is positive or negative with explanation.
From tip, it is clear that, as the price increases, demand goes down.
So, there is inverse relation between price and demand.
Hence, demand will have negative slope.
If the horizontal axis represents price, and the vertical axis represents quantity, demand will have negative slope.
Page 107 Problem 5 Answer
A company produces a security device known as Toejack.
Toejack is a computer chip that parents attach between the toes of a child, so parents can track the child’s location at any time using an online system.
The company has entered into an agreement with an Internet service provider, so the price of the chip will be low.
The company decides to conduct a market research survey to determine the best price for the device.
The variable p represents price, and q represents quantity demanded. The points are listed as(p,q)
(14,8200),(11,9100),(16,7750),(16,8300),(14,8900)
(17.7100).(13.8955),(11.9875),(11.9425).(18.5825)
To make a scatter plot of the data ad to check if the data look like it has a linear form.
The graph is
It almost seem to be a straight line.
The scatter plot of the data is
The data seem linear form.
Page 107 Problem 6 Answer
A company produces a security device known as Toejack.
Toejack is a computer chip that parents attach between the toes of a child, so parents can track the child’s location at any time using an online system.
The company has entered into an agreement with an Internet service provider, so the price of the chip will be low.
To check if the linear regression line a good predictor and explain.
We will draw graph.
The graph is
It is a good predictor.
The regression line is a good predictor.
Page 107 Problem 7 Answer
A company produces a security device known as Toejack.
Toejack is a computer chip that parents attach between the toes of a child, so parents can track the child’s location at any time using an online system.
The company has entered into an agreement with an Internet service provider, so the price of the chip will be low.
To examine the data to see if there is any relationship between the price and the quantity demanded.
To determine the correlation coefficient between price and demand, rounded to nearest hundredth.
To explain the significance of the correlation coefficient.
We will calculate the correlation coefficient using machine.
The correlation coefficient is−0.9156370841≈−0.92
As the coefficient is negative, there is a negative relation between price and quantity demanded.
Also, since the coefficient is almost one in magnitude, the relation is strong.
Correlation coefficient between price and demand is≈−0.92
It shows there is a negative relation which is strong too.
Page 107 Problem 8 Answer
A company produces a security device known as Toejack.
Toejack is a computer chip that parents attach between the toes of a child, so parents can track the child’s location at any time using an online system.
The company has entered into an agreement with an Internet service provider, so the price of the chip will be low.
Fixed cost $24500 variable cost per variable $6.12
To express expense E as a function of q demand.
The variable cost will be6.12q
So, E=6.12q+15400
The expense equation E as a function of demand q is
E=612q+15400.
Page 107 Problem 9 Answer
A company produces a security device known as Toejack.
Toejack is a computer chip that parents attach between the toes of a child, so parents can track the child’s location at any time using an online system.
The company has entered into an agreement with an Internet service provider, so the price of the chip will be low.
To express revenue R in terms of p,q
From tip,
R=pq
Revenue R in terms of p,q is R=pq.
Page 107 Problem 10 Answer
A company produces a security device known as Toejack.
Toejack is a computer chip that parents attach between the toes of a child, so parents can track the child’s location at any time using an online system.
The company has entered into an agreement with an Internet service provider, so the price of the chip will be low.
To use he transitive property of dependence to express expense, E, in terms of p. Round to the nearest hundredth.
From before,
E=6.12q+24500
q=−423.61p+14315.94
E=−2592.49p+112113.55
Expense E in terms of p is
E=−2592.49p+112113.55.
Page 107 Exercise 1 Answer
A company produces a security device known as Toejack.
Toejack is a computer chip that parents attach between the toes of a child, so parents can track the child’s location at any time using an online system.
The company has entered into an agreement with an Internet service provider, so the price of the chip will be low.
To determine an appropriate maximum horizontal-axis value.
The function E is E=−2592.49p+112113.55
From tip, E=0
p≈44
The maximal horizontal value is 44.
Page 107 Exercise 2 Answer
A company produces a security device known as Toejack.
Toejack is a computer chip that parents attach between the toes of a child, so parents can track the child’s location at any time using an online system.
The company has entered into an agreement with an Internet service provider, so the price of the chip will be low.
To determine an appropriate maximum vertical-axis value.
The maximal vertical axis value will be the Expense at price zero
Which is112113.55≈113000
The maximal vertical-axis value is 113000.
Page 107 Exercise 3 Answer
A company produces a security device known as Toejack.
Toejack is a computer chip that parents attach between the toes of a child, so parents can track the child’s location at any time using an online system.
The company has entered into an agreement with an Internet service provider, so the price of the chip will be low.
To sketch the graphs of the expense and revenue functions.
The graph is
The graph between revenue function and expense is
Page 107 Exercise 4 Answer
A company produces a security device known as Toejack.
Toejack is a computer chip that parents attach between the toes of a child, so parents can track the child’s location at any time using an online system.
The company has entered into an agreement with an Internet service provider, so the price of the chip will be low.
Determine the coordinates of the maximum point on the revenue graph. Round the coordinates to the nearest hundredth.
The revenue function is
R=−423.61p2+14315.94p
From tip, Maximum occurs at
p≈16.9
R≈120952.14
The coordinates of the maximum point on the revenue graph is(16.9,120952.14).
Page 107 Exercise 5 Answer
A company produces a security device known as Toejack.
Toejack is a computer chip that parents attach between the toes of a child, so parents can track the child’s location at any time using an online system.
The company has entered into an agreement with an Internet service provider, so the price of the chip will be low.
To determine the breakeven points.
The Expense function is
E=−2,592.49p+112,113.55
Revenue function is
R=−423.61p{2}+14,315.94p
−423.61p{2}+14,315.94p=−2,592.49p+112,113.55
p≈31.52,8.40
Points are 31.52,30403.76
8.40,90343.87
The breakeven points are 31.52,30403.76
8.40,90343.87
Page 107 Exercise 6 Answer
Here we have
Revenue function found in exercise 12 isR=−423.61p2+14,315.94p
Expense function found in exercise 13 is E=−2,592.49p+112,113.55
By using these functions we will find the profit function.
Profit function is the difference between the revenue function and the expense function
Such thatP=R−E
=−423.61p2+16,908.43p−112,113.55
Thus profit function P, in terms of p. is −423.61p2+16,908.43p−112,113.55
Page 107 Exercise 7 Answer
We have profit functionP=−423.61p2+16,908.43p−112,113.55
Here first we will value of price at maximum revenue happen and that we will draw the graph.
We have Profit functionP=−423.61p2+16,908.43p−112,113.55
The maximum Profit occurs at the axis of symmetry x=−b/2a for a quadratic function y=ax2+bx+c
Thus p=−16,908.43
2(−423.61)
≈19.96
Thus the maximum profit happens at a price of $19.90.
Graphical representation
The maximum profit happens at a price of $19.90.
Page 107 Exercise 8 Answer
We haveq=−423.61p+14,315.94
R=−423.61p2+14,315.94p
E=−2,592.49p+112,113.55
P=−423.61p2+16,908.43p−112,113.55
Break-even points are at prices of $8.40 and $31.52 (result exercise 16 ).
The maximum profit price is $19.96
Here by using these results we will answer the given statements.
(1) q Toejacks should be manufactured (produeed) at a price of $19.96 each.
q=−423.61(19.96)+14,315.94=5860.6844≈5861
(2) Every Toejack will be sold at a price of $19.96 per Toejack, which is the maximum profit price.
(3) & (4) The breakeven points are reached at a price of $8.40 or 831.52
(5) The revenue is given by R=−423.61p2+14,315.94p at p=19.96
R=−423.61(19.96)2+14,315.94(19.96)=116979.260624≈8116979.26
(6) The expenses are given by E=−2,592.49p+112,113.55 at p=19.96
E=−2,592.49(19.96)+112,113.55=60367.4496≈$60367.45
(7) The profit is given by P=−423.61p2+16,908.43p−112,113.55 at p=19.96
P=−423.61(19.96)2 +16,908.43(19.96)−112,113.55=56611.811024≈$56611.81
Answers are (1)5861(2)$19.96(3)$8.40(4)$31.52(5)$116,979.26(6)$60,367.45(7)$56,611.81
Page 107 Exercise 9 Answer
We have E=−2,592.49p+112,113.55
The maximum profit was obtained at a price of $19.90 as derived in a previous exercise.
Assume that the business holders decide to use this price, then we can determine the corresponding
Expenses: E=−2,592.49(19.96)+112,113.55=60367.4496 Thus the total expenses are then $00,367.4490 or approximately $00,307.45
The shares are sold at $5 per share. Let there be x shares, then the total income of the shares is 5x.
The income of the shares should cover the total expenses: 5x=$60,307.45
Thus we obtain x=60,367.45/5
=12,073.49
Thus 12,073.49 shares should be sold.
However, the number of shares should be an integer and thus we should sell at least 12,074 shares to get enough money to start the business.
Number of shares must be sold to get enough money to start the business is 12074.
Chapter 2 Solving Linear Inequalities
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- Cengage Financial Algebra 1st Edition Chapter 2 Assessment Modeling a Business
- Cengage Financial Algebra 1st Edition Chapter 2 Exercise 2.1 Modeling a Business
- Cengage Financial Algebra 1st Edition Chapter 2 Exercise 2.2 Modeling a Business
- Cengage Financial Algebra 1st Edition Chapter 2 Exercise 2.3 Modeling a Business
- Cengage Financial Algebra 1st Edition Chapter 2 Exercise 2.4 Modeling a Business
- Cengage Financial Algebra 1st Edition Chapter 2 Exercise 2.5 Modeling a Business
- Cengage Financial Algebra 1st Edition Chapter 2 Exercise 2.6 Modeling a Business
- Cengage Financial Algebra 1st Edition Chapter 2 Exercise 2.7 Modeling a Business