Q: Which of the following best describes the relationship between profit, revenue, and cost?

$$\text{Profit} = \text{Revenue} - \text{Cost}$$

Question 1

A company sells $$x$$ units of a product per week. The price-demand equation is $$p(x) = 400 - 0.4x$$ and the cost function is $$C(x) = 2000 + 160x. $$What is the formula for the weekly profit $$P(x)$$?

Accepted Answer

$$P(x) = 400x - 0.4x^2$ - 2000 - 160x$$

Question 2

Given the price-demand equation $$p(x) = 400 - 0.4x$$, what is the number of cameras $$x$$ that maximizes weekly revenue?

Accepted Answer

$$x = 500$$

Question 3

For the same company, what price should be charged to maximize weekly revenue?

Accepted Answer

$$p = 200$$

Question 4

What is the maximum weekly profit for the company, given $$p(x) = 400 - 0.4x$$ and $$C(x) = 2000 + 160x$$?

Accepted Answer

$$34,000$$

Question 5

Which of the following is the correct sequence of steps in the general optimization method described in the study materials?

Accepted Answer

Identify the quantity to be optimized, find a formula, express in one variable, find critical numbers, verify maximum/minimum, answer the question

Question 6

If two numbers have a sum of 21, what are the values that maximize their product?

Accepted Answer

$$10.5$$ and $$10.5$$

Question 7

Find two positive numbers whose product is 21 and whose sum is minimized.

Accepted Answer

$$\sqrt{21}$$ and $$\sqrt{21}$$

Question 8

A rectangle has an area of $$36$$ square feet. What are the dimensions that minimize its perimeter?

Accepted Answer

$$6\sqrt{3} ft by$$ $$6\sqrt{3} ft$$

Question 9

A rectangle has a perimeter of $$76$$ feet. What are the dimensions that maximize its area?

Accepted Answer

$$19 ft by$$ $$19 ft$$

Question 10

A box is to be made from a $$12 by$$ $$12$$ inch piece of cardboard by cutting equal squares from each corner and folding up the sides. What size square should be cut to maximize the volume?

Accepted Answer

$$2$$ inches

Question 11

Suppose the cost function for producing $$x$$ items is $$C(x) = 2000 + 160x$$ and the price-demand function is $$p(x) = 400 - 0.4x. $$What is the revenue function $$R(x)$$?

Accepted Answer

$$R(x) = x(400 - 0.4x)$$

Question 12

In a business optimization problem, what does the critical number of $$Q(x)$$ represent?

Accepted Answer

A candidate for the value of $$x$$ that maximizes or minimizes $$Q(x)$$

Question 13

Which of the following best describes the relationship between profit, revenue, and cost?

Accepted Answer

$$	ext{Profit} = 	ext{Revenue} - 	ext{Cost}$$

Question 14

A company wants to maximize its profit. Which step should be performed first according to the general method of attack?

Accepted Answer

Identify the quantity to be optimized

Question 15

If the price-demand equation is $$p(x) = 400 - 0.4x$$, what is the maximum possible revenue?

Accepted Answer

$$100,000$$

Question 16

Which constraint must be considered when expressing the quantity to be optimized in terms of one variable?

Accepted Answer

Any equations or relationships given in the problem

Question 17

A company finds that producing 300 cameras and charging $280 per camera yields the maximum profit. What is the total revenue at this point?

Accepted Answer

$$84,000$$

Question 18

If a rectangle has a fixed area, what shape minimizes its perimeter?

Accepted Answer

A square

Question 19

If a rectangle has a fixed perimeter, what shape maximizes its area?

Accepted Answer

A square

Question 20

When constructing a box from a square piece of cardboard by cutting equal squares from each corner, what is the general approach to maximize the volume?

Accepted Answer

Express the volume as a function of the side of the cut square, find the critical number, and verify it gives a maximum

Optimization in Business Calculus: Methods and Applications

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