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Math 1064 Problem Solving Assignment 3 Sections 1.7 and 3.4

Problem Solving Assignment 3

In this assignment, we will work on solving the following

Problem 1: Imagine you are the owner and the CEO of a small company manufac-

turing specialty kayaks. You have designed a new kayak and are trying to analyze

the costs, the revenues, the profits associated with the production and sales of this

new product. You have determined that you will have to spend $150,000 on the new

production line set-up and the advertising, and that after that it will cost you $400

to produce each kayak. You also conducted market research which showed that the

demand equation for the new kayak is

p + 10q = 3000 (1)

where q is the number of the kayaks that will be purchased at the price p dollars

per kayak. You decided that you will always price the kayaks at the highest price at

which all manufactured kayaks will be sold. Now, do the following further analysis

1) Express the company’s costs C that will be incurred in the production of the

new kayak, the revenue R and the profit P from the sales of the kayak as

functions of the quantity q of these kayaks that are manufactured and sold

2) Determine what level(s) of production (the number of kayaks manufactured)

will result in the company breaking even

3) Determine what level(s) of production will lead to the company making as large

profit as possible; what is the maximum possible profit that the company can

make by manufacturing and selling the new kayak?

4) Determine what level(s) of production will result in the company’s revenue being

as large as possible?

Step 1. Understanding the problem: Analyzing the costs

Calculate how much it will cost the company to produce just one new kayak,

two kayaks, ten kayaks, q kayaks

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Math 1064 Problem Solving Assignment 3 Sections 1.7 and 3.4

Step 2. Understanding the problem: the Demand equation.

Using the demand equation (1), answer the following questions:

1. If the kayaks are priced at $1200 each, how many kayaks could be sold?

2. If the kayaks are given away for free and there is an unlimited supply of

these kayaks, how many will be claimed?

3. At what price levels the company will not be able to sell even one kayak?

4. If the company offers the kayaks at $800 each and has a supply of 200

kayaks, what will happen?

A. Some kayaks will remain unsold

B. The company will run out of kayaks before all customer’s demand will

be met

C. The company will have just enough kayaks to satisfy all customers

who would like to purchase it at this price

5. Discuss with your teammates what the sentence “You decided that you

will always price the kayaks at the highest price at which all manufactured

kayaks will be sold” in the statement of the problem means. How do you

understand it?

6. If 100 kayaks have been manufactured, what is the highest price the com-

pany can charge for one kayak in order to sell all 100?

Step 3. Understanding the problem: Analyzing the Revenue.

Let’s remember that the company’s revenue R from the sales of the new kayak

can be determined as

R = Price per kayak × Number of kayaks sold (2)

Suppose the company manufactured q kayaks and priced them so that 1) all

q kayaks will be sold and 2) the company’s revenue is as high as possible.

Following up on the discussion in the previous step, express the price p per

kayak in terms of q. Now, substitute this expression for p in terms of q into the

above formula (2). You expressed the company’s revenue R as a function of q!

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Math 1064 Problem Solving Assignment 3 Sections 1.7 and 3.4

Step 4. Understanding the problem: Analyzing the Profit. Use the formula

for the profit

Profit = Revenue − Cost (3)

and your results so far, to express the profit P as a function of the number of

kayaks q that are manufactured.

Step 5. Solving the Problem: How to Break Even

The company breaks even when the revenue is just high enough to cover the

costs. In other words, when R = C and P = R−C = 0. In the previous steps, you expressed the profit P as a function of q obtaining an expression P (q) for

P : P = P (q). Now, solve the equation

P (q) = 0 (4)

to find the level(s) of production at which the company will break even.

Step 6. Solving the Problem: Maximizing the Profit

In the previous steps, you expressed the profit P as a function of the production

level q. What kind of a function is it? Without actually graphing it, think about

the shape of its graph. For a graph of that shape, where would the highest point

on the graph be located? Returning to the equation P = P (q), can you find

the value(s) of q that would produce the highest value of P? Once you found

this value of q, think about how you could find the maximum possible profit

(i.e. the maximum possible value of P ) itself.

Step 7. Solving the Problem: Maximizing the Revenue

Repeat Step 6, replacing the profit function P = P (q) with the revenue function

R = R(q).

Now, solve the problems below. If you are having difficulties, try to think if you

could use the steps above, possibly modified to suit a new problem, to make

progress.

Problem 2: A company manufacturing vacuum cleaners analyzed its cost and

researched the market conditions. The results of this research were as follows:

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Math 1064 Problem Solving Assignment 3 Sections 1.7 and 3.4

the company’s fixed costs were $161,250, the marginal cost was $300 per vacuum

cleaner, while the demand equation for the vacuum cleaners was

p + 2q = 2000

where q is the number of the vacuum cleaners that will be purchased at the

price p dollars per vacuum cleaner. Assume that the company always prices

the vacuum cleaners at the highest price at which all manufactured vacuum

cleaners will be sold. Now, complete further analysis as requested below

1) Express the company’s costs C, the revenue R and the profit P as functions

of the quantity q of the vacuum cleaners that are manufactured and sold

2) Determine what level(s) of production will result in the company breaking

even

3) Determine what level(s) of production will lead to the company making

as large profit as possible; what is the maximum possible profit that the

company can make by manufacturing and selling the vacuum cleaners?

4) Determine what level(s) of production will result in the company’s revenue

being as large as possible?

Problem 3: There are 42 apple trees in an orchard. Each tree produces 600

apples. For each additional tree planted in the orchard, the output per tree

drops by 10 apples (because the trees are more crowded). How many trees

should be added to the existing orchard in order to maximize the number of

apples produced by all trees in the orchard?

Answer: 9

Problem 4: Jessica is selling freshly squeezed orange juice. It costs her 20¢ to

make each glass of juice. If she charges 60¢ per glass, then 70 people will buy

a glass. If she charges $2.00 per glass, then only 35 people will buy a glass.

Assume the number of people who will buy orange juice is a linear function of

the price (in other words, the demand equation is a linear equation). What

should Jessica charge for a glass of juice to maximize her profit?

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