Question

The amount of revenue Dave makes can be modeled by the equation R = (p − 1)(200 − 40p),where p represents the price of one donut and R represents the total revenue. Use as many strategies asyou can to figure out the questions below.


1.At what price will Dave break even (make no revenue)?

2. How much should Dave charge to maximize his revenue?

3.What is the most money Dave can make?

4.Dave made $100 from donut sales. How much must he have charged per donut?

Please Explain

Answer 1

The solution to the questions are:

• Dave's break even (make no revenue) at prices of 1 and 5
• Dave should charge a price of 3 to maximize his revenue
• The most money Dave can make is a revenue of $160
• The prices per donut to make a revenue of $100 are $0.13 and $5.87

1. At what price will Dave break even (make no revenue)?

The revenue function is given as:

R = (p − 1)(200 − 40p)

At break even, R = 0

So, we have:

(p − 1)(200 − 40p) = 0

Split the above equation

p − 1 = 0 and 200 − 40p = 0

Solve for P

P = 1 and P = 5

Hence, Dave break even (make no revenue) at prices of 1 and 5

2. How much should Dave charge to maximize his revenue?

Recall that:

R = (p − 1)(200 − 40p)

Expand the factors

R = 200p - 40p^2 - 200 + 40p

Differentiate the above equation

R' = 200 - 80p + 40

Set the equation to 0

200 - 80p + 40 = 0

Evaluate the like terms

-80p = -240

Divide by -80

p = 3

Hence, Dave should charge a price of 3 to maximize his revenue

3. What is the most money Dave can make?

In (2), we have:

p = 3

Substitute p = 3 in R = (p − 1)(200 − 40p)

R = (3 − 1)(200 − 40 * 3)

Evaluate

R = 160

Hence, the most money Dave can make is a revenue of $160

4. Dave made $100 from donut sales. How much must he have charged per donut?

This means that:

R = $100

So, we have:

100 = (p − 1)(200 − 40p)

Expand the factors

100 = 200p - 40p^2 - 200 + 40p

Evaluate the like terms

40p^2 - 240p + 300 = 0

Divide through by 20

20p^2 - 120p + 150 = 0

Solve for p

p = 0.13 and p = 5.87

Hence, the prices per donut to make a revenue of $100 are $0.13 and $5.87

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