Q: A company wants to build a rectangular enclosure using 200 ft of fencing, with one side formed by a rock wall that does not require fencing. What are the dimensions that maximize the area of the enclosure?

$$50 \text{ ft} \times 75 \text{ ft}$$

Q: A box with a square base and no lid must have a volume of $$125\ \text{in}^3. $$What should the dimensions be to minimize the amount of cardboard used?

Base: $$5\ \text{in}$$, Height: $$5\ \text{in}$$

Q: A poster must have a total area of $$1150\ \text{cm}^2$$, with $$2\ \text{cm}$$ margins on the sides and top, and a $$3\ \text{cm}$$ margin at the bottom. What dimensions maximize the printed area?

Width: $$25\ \text{cm}$$, Height: $$40\ \text{cm}$$

Q: A 10-meter-tall tree stands 5 meters from a cliff. What is the length of the shortest rope that can be tied from the ground over the top of the tree to the top of the cliff?

$$\sqrt{5^2$ + (h+10)^2$}$$,\ h=\text{height of cliff}$

Question 1

A company wants to build a rectangular enclosure using 200 ft of fencing, with one side formed by a rock wall that does not require fencing. What are the dimensions that maximize the area of the enclosure?

Accepted Answer

$$50 	ext{ ft} 	imes 75 	ext{ ft}$$

Question 2

A box with a square base and no lid must have a volume of $$125\ 	ext{in}^3. $$What should the dimensions be to minimize the amount of cardboard used?

Accepted Answer

Base: $$5\ 	ext{in}$$, Height: $$5\ 	ext{in}$$

Question 3

A poster must have a total area of $$1150\ 	ext{cm}^2$$, with $$2\ 	ext{cm}$$ margins on the sides and top, and a $$3\ 	ext{cm}$$ margin at the bottom. What dimensions maximize the printed area?

Accepted Answer

Width: $$25\ 	ext{cm}$$, Height: $$40\ 	ext{cm}$$

Question 4

A coffee shop sells $$x$$ coffees per day at a price $$p(x) = 100 - \frac{x}{4}. $$What value of $$x$$ maximizes daily revenue?

Accepted Answer

$$x = 400$$

Question 5

Gerald and Katya sell custom phone cases. The price-demand function is $$p(x) = 50 - 0.2x$$ and the cost function is $$C(x) = 100 + 15x. $$What is the number of cases $$x$$ that maximizes profit?

Accepted Answer

$$x = 87.5$$

Question 6

A math store sells 400 calculators per year. It costs $$2 to $$store one calculator for a year, and each order costs $$25. $$How many calculators should be ordered each time to minimize total costs?

Accepted Answer

$$63$$

Question 7

A Norman window has a perimeter of $$24$$ meters. The semicircular part lets in one quarter as much light per square meter as the rectangular part. What width $$x$$ maximizes the total light entering?

Accepted Answer

$$x = 6$$

Question 8

A box with an open top is formed by cutting equal squares from the corners of a $$3\ 	ext{ft} 	imes 8\ 	ext{ft}$$ cardboard and folding up the sides. What value of $$x ($$side of the square) maximizes the volume?

Accepted Answer

$$x = 1.5$$

Question 9

A 10-meter-tall tree stands 5 meters from a cliff. What is the length of the shortest rope that can be tied from the ground over the top of the tree to the top of the cliff?

Accepted Answer

$$\sqrt{5^2$ + (h+10)^2$}$$,\ h=	ext{height of cliff}$

Question 10

Find $$\frac{dy}{dx}$$ for $$5x^2$$ + $$y^3 = 12$ using implicit differentiation.

Accepted Answer

$$\frac{dy}{dx} = -\frac{10x}{3y^2$}$$

Question 11

Given $$x^2$$ + $$y^2 = 49$, find $$\frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

$$\frac{dy}{dx} = -\frac{x}{y}$$

Question 12

A sphere is growing at a rate of $$50\ 	ext{cm}^3/	ext{s}. At $$what rate is the radius increasing when the radius is $$5\ 	ext{cm}$$?

Accepted Answer

$$\frac{dr}{dt} = \frac{50}{4\pi (5^2$)}$$

Question 13

A right triangle has a base of $$10\ 	ext{cm}$$ and a height of $$12\ 	ext{cm}. If $$the height is decreasing at $$0.4\ 	ext{cm}/	ext{s}$$, at what rate is the area decreasing?

Accepted Answer

$$-2\ 	ext{cm}^2/	ext{s}$$

Question 14

If $$f(x) = x^3$ + x$$, find the differential $$dy$$ when $$x = 2$$ and $$dx = 0.1.$$

Accepted Answer

$$dy = 13 	imes 0.1 = 1.3$$

Question 15

Given $$f(x) = x^3$ + x$$, use differentials to estimate $$f(2.1).$$

Accepted Answer

$$f(2.1) \approx 11.3$$

Question 16

If $$f(x) = x^3$ + 1$$, use the linearization at $$a = 5 to $$approximate $$f(5.1).$$

Accepted Answer

$$f(5.1) \approx 126.51$$

Question 17

If $$f(x) = \sqrt{x + 12}$$, use the linearization at $$a = 16 to $$approximate $$f(16.01).$$

Accepted Answer

$$f(16.01) \approx 4.00125$$

Applications of the Derivative: Optimization, Implicit Differentiation, Related Rates, Differentials, and Linearization

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