Question 1

A company wants to maximize the area of a rectangular garden using 100 meters of fencing. One side of the garden will be against a wall and does not require fencing. What are the dimensions of the garden that maximize the area?

Accepted Answer

Question 2

Given the cost function $$C(x) = x^3$ - 6x^2$ + 13x + 15$$, what is the value of $$x$$ that minimizes the marginal cost?

Accepted Answer

$$x = 2$$

Question 3

A rectangular pen is to be built with three parallel walls using 500 feet of fencing. What are the dimensions that maximize the total area?

Accepted Answer

Width $$= 83.33 ft$$, Length $$= 166.67 ft$$

Question 4

The demand equation for a product is $$p(x) = -2x + 55$$ and the cost function is $$C(x) = 21x + 98. $$What is the maximum profit?

Accepted Answer

$$P_{max} = 252$$

Question 5

A local office supply store has an annual demand for 50,000 cases of paper. It costs $$4 per year to store a case and 50 to $$place an order. What is the optimal order quantity (Economic Order Quantity, EOQ)?

Accepted Answer

$$1,000$$ cases

Question 6

The revenue function for a firm is $$R(x) = 200x - 1600x/(x+8) - x^2$. What value of x$ maximizes the revenue?

Accepted Answer

$$x = 16$$

Question 7

A builder wants to construct a rectangular pen with three parallel walls using 500 feet of fencing. If $$x is $$the width and $$y is $$the length, which equation represents the constraint?

Accepted Answer

$$2y + 3x = 500$$

Question 8

If the price of a funnel cake is $$p$$ and the number sold is $$x$$, and the demand is linear, which of the following is the correct form for the demand equation?

Accepted Answer

$$p = mx + b$$

Question 9

A farmer has $$750 to build an E-shaped fence along a river to create two identical rectangular pastures. The side parallel to the river costs 6/ft$, and the three sections perpendicular to the river cost 5/ft$. If x$ is the length parallel to the river and y$ is the width, which equation represents the cost constraint?

Accepted Answer

$$6x + 3 \cdot 5y = 750$$

Question 10

Given the demand equation $$p(x) = \frac{1}{12}x^2$ - 10x + 300$$ for $$0 \leq x \leq 60$$, what value of $$x$$ maximizes the revenue?

Accepted Answer

$$x = 48$$

Question 11

When the average ticket price for a concert was $$50, attendance was 4000. When the price was raised to 52$$, attendance dropped to 3800. Assuming a linear demand, what ticket price maximizes revenue?

Accepted Answer

$$p = 51$$

Question 12

If the cost function is $$C(x) = 2x^2$ + 5x + 100$$ and the revenue function is $$R(x) = 25x$$, what is the profit-maximizing quantity?

Accepted Answer

$$x = 10$$

Question 13

A company sells $$x$$ units at price $$p(x) = 100 - 2x. $$What is the revenue function $$R(x)$$?

Accepted Answer

$$R(x) = 100x - 2x^2$

Question 14

A rectangular garden is to be fenced on three sides with $$320$$ dollars available. The side facing the street costs $$6/ft$$ and the other three sides cost $$2/ft. If$$ $$x is $$the length parallel to the street and $$y is $$the width, which equation represents the cost constraint?

Accepted Answer

$$6x + 2 \cdot 2y = 320$$

Question 15

If the marginal cost is the derivative of the cost function, what is the marginal cost for $$C(x) = 3x^2$ + 2x + 7 at$$ $$x = 4$$?

Accepted Answer

$$MC = 31$$

Question 16

If the cost to store a case of paper is $$4$$ per year and the cost to place an order is $$50$$, with an annual demand of $$50,000$$ cases, what is the formula for the total inventory cost $$C(Q) as a $$function of order quantity $$Q$$?

Accepted Answer

$$C(Q) = \frac{50,000}{Q} \cdot 50 + \frac{Q}{2} \cdot 4$$

Question 17

Sherwood Stadium sells funnel cakes. When the price is $$5.90$$, 2750 are sold; when the price is $$6.20$$, 2000 are sold. Assuming a linear demand, what is the slope of the demand equation?

Accepted Answer

$$m = -0.00048$$

Question 18

If the profit function is $$P(x) = -2x^2$ + 34x - 98$$, what is the maximum profit and at what $$x$$ does it occur?

Accepted Answer

Maximum profit at $$x = 8.5$$, $$P(8.5) = 47$$

Question 19

If the revenue function is $$R(x) = x \cdot p(x)$$ and $$p(x) is $$the demand equation, what does the maximum of $$R(x)$$ represent in a business context?

Accepted Answer

The maximum possible revenue from sales

Optimization Problems and Applications of Derivatives in Business Calculus

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