Q: A company wants to build a rectangular enclosure using 100 meters of fencing for three sides (the fourth side is a river and needs no fence). What should the lengths of the sides be to maximize the area, and what is the maximum area?

$$x = 25 m$$, $$y = 50 m$$; \text{max area} = 1250$$ m^2$

Question 1

Which of the following best describes a 'critical number' of a function $$f(x)$$?

Accepted Answer

A value of $$x$$ where $$f'(x) = 0 or$$ $$f'(x) is $$undefined.

Question 2

A company wants to build a rectangular enclosure using 100 meters of fencing for three sides (the fourth side is a river and needs no fence). What should the lengths of the sides be to maximize the area, and what is the maximum area?

Accepted Answer

$$x = 25 m$$, $$y = 50 m$$; 	ext{max area} = 1250$$ m^2$

Question 3

Given the function $$f(x) = -x^2$ + 4x + 1 on $$the interval $$[0, 5]$$, what is the absolute maximum value and where does it occur?

Accepted Answer

At $$x = 2$$, $$f(2) = 5$$

Question 4

A rectangle is inscribed in the first quadrant under the line $$y = -\frac{1}{2}x + 6. $$What is the maximum possible area of such a rectangle?

Accepted Answer

$$18$$ units$$^2$$

Question 5

Which of the following is the correct procedure for solving an applied optimization problem?

Accepted Answer

Draw a picture, label variables, use the constraint to reduce to one variable, determine the domain, differentiate, find and classify critical numbers, and check endpoints if needed.

Question 6

A company wants to design an open-top cylindrical can to hold $$100\pi in$^3 of $$liquid. What should the radius $$r$$ and height $$h be to $$minimize the surface area?

Accepted Answer

$$r = \sqrt[3]{100}$$, $$h = \sqrt[3]{100}$$

Question 7

Suppose you are minimizing the sum $$S = x + y$$ given the constraint $$xy = 100$$ for $$x, y \u003e 0. $$What is the minimum value of $$S$$?

Accepted Answer

$$S = 20$$

Question 8

Which test would be most efficient for classifying a critical point if the second derivative is easy to compute?

Accepted Answer

Second Derivative Test

Question 9

A garden of area $$150 yd$^2 is to be $$surrounded by sidewalks of width $$1 yd on $$the west, north, and east sides, and $$2 yd on $$the south side. What is the correct expression for the total area (garden plus sidewalks) in terms of the garden width $$x$$?

Accepted Answer

$$T(x) = (x + 2)(y + 3)$$ where $$y = 150/x$$

Question 10

You are asked to find the point on the line $$y = \frac{2}{3}x - 5$$ that is closest to the point $$(5, 7). $$Which function should you minimize?

Accepted Answer

$$D^2(x) = (x$$ - $$5)^2$$ + (\frac{2}{3}x - 5 - $$7)^2$$

Question 11

If $$f''(c) = 0 at a $$critical number $$c$$, what should you do to classify the extremum?

Accepted Answer

Use the first derivative test.

Question 12

A rectangular pen is to be built against a wall using $$72 ft of $$fencing for the other three sides. What is the maximum area that can be enclosed?

Accepted Answer

$$648 ft$^2$$

Question 13

Which of the following is NOT a step in the standard procedure for solving an applied optimization problem?

Accepted Answer

Differentiate the constraint equation.

Question 14

For the function $$f(x) = x^3$ - 3x^2$ + 2 on$$ $$[0, 3]$$, what is the absolute minimum value?

Accepted Answer

$$f(2) = -2$$

Question 15

Why is it often preferable to minimize the squared distance rather than the distance itself in optimization problems?

Accepted Answer

Because it avoids the square root, making differentiation easier.

Question 16

Given $$S(y) = \frac{100}{y} + y$$ for $$y \u003e 0$$, what is the value of $$y$$ that minimizes $$S$$?

Accepted Answer

$$y = 10$$

Question 17

If you are maximizing a function $$f(x) on a $$closed interval $$[a, b]$$, which values must you check to find the absolute maximum?

Accepted Answer

Both the critical numbers in $$[a, b]$$ and the endpoints.

Question 18

A rectangle is inscribed under the line $$y = -\frac{1}{2}x + 6 in $$the first quadrant. What are the dimensions of the rectangle with maximum area?

Accepted Answer

$$x = 6$$, $$y = 3$$

Question 19

Which of the following best describes the 'constraint' in an optimization problem?

Accepted Answer

The equation that variables must satisfy exactly.

Optimization Problems in Calculus: Strategies, Tests, and Applications

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