Q: Which of the following is the correct objective equation for maximizing the area $$A of a $$rectangular garden with one side against a house and $$P$$ feet of fencing for the other three sides?

$$A = x \times y$$ where $$2x + y = P$$

Question 1

A homeowner wants to build a rectangular garden along one side of their house, using the house as one side and fencing for the other three sides. If they have 100 feet of fencing available, what dimensions will maximize the area of the garden?

Accepted Answer

Width $$= 25 ft$$, Length $$= 50 ft$$

Question 2

Which of the following is the correct objective equation for maximizing the area $$A of a $$rectangular garden with one side against a house and $$P$$ feet of fencing for the other three sides?

Accepted Answer

$$A = x 	imes y$$ where $$2x + y = P$$

Question 3

A ball is thrown straight up with an initial velocity of $$48 ft/s. $$Its height after $$t$$ seconds is given by $$h(t) = 48t - 16t^2$. At what time does the ball reach its maximum height?

Accepted Answer

$$t = 1.5$$ seconds

Question 4

Given the cost function for building a rectangular enclosure with three sides of redwood fencing at $$\14$ per foot and one side of cement blocks at $$\$$28$$ per foot, and an area of $$600 ft$^2$$, which equation represents the total cost $$C in $$terms of $$x ($$width) and $$y ($$length)?

Accepted Answer

$$C = 28x + 42y$$

Question 5

A manufacturer wants to minimize the cost of maintaining and replacing a machine. Which calculus concept is most directly used to find the optimal replacement interval?

Accepted Answer

Finding the minimum of a cost function using derivatives

Question 6

Suppose you want to maximize the volume $$V of a $$box with a square base and no top, using $$96 ft$^2 of $$material for the sides and base. Which constraint equation should you use?

Accepted Answer

$$x^2 + 4xh = 96$

Question 7

A department store manager wants to build a $$600 ft$^2$$ rectangular enclosure with three sides of redwood fencing and one side of cement blocks. If $$x is $$the width and $$y is $$the length, what is the constraint equation?

Accepted Answer

$$xy = 600$$

Question 8

For the function $$f(x) = -16x^2$ + 48x + 4$$, what is the maximum value of $$f(x)$$?

Accepted Answer

$$40$$

Question 9

Which of the following steps is NOT part of the standard process for solving optimization problems?

Accepted Answer

Integrate the objective function over the constraint interval

Question 10

A canvas wind shelter for the beach has a back, two sides, and a top. If $$96 ft of $$canvas is to be used, and the shelter has width $$x$$ and height $$h$$, what is the equation for the total canvas used?

Accepted Answer

$$x h + 2 h x + x^2$ = 96$$

Question 11

If you want to maximize the area $$A of a $$rectangular garden with one side against a house and $$P$$ feet of fencing for the other three sides, what is the value of $$x ($$the width perpendicular to the house) that maximizes the area?

Accepted Answer

$$x = \frac{P}{4}$$

Question 12

A doctor wants to find the minimum amount of a drug that will produce a desired response in a patient. Which mathematical process should be used?

Accepted Answer

Minimize the dosage function using derivatives

Question 13

Which test can be used to confirm whether a critical point found in an optimization problem is a maximum or minimum?

Accepted Answer

Second derivative test

Question 14

A rectangular field is to be fenced on three sides with 200 ft of fencing, with one side against a barn. What is the maximum area that can be enclosed?

Accepted Answer

$$5,000 ft$^2$$

Question 15

If the cost to build a rectangular enclosure is $$C = 28x + 42y$$ and the area is $$xy = 600$$, what is the minimum cost?

Accepted Answer

$$1,680$$

Optimization Problems in Business Calculus: Applications of the Derivative

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