Q: A closed rectangular box has a volume of $$972\ \mathrm{cm}^3$$ and a width three times its length. Which set of dimensions minimizes the surface area?

$$6\ \mathrm{cm} \times 18\ \mathrm{cm} \times 9\ \mathrm{cm}$$

Question 1

A farmer wants to build a rectangular pen divided into two equal rectangular sections. The outer wall costs $$5 per foot, and the inner dividing wall costs 3 $$per foot. If the total area must be 2600 sq ft, which of the following expressions gives the total cost $$C in $$terms of the pen's length $$x$$ and width $$y$$?

Accepted Answer

$$C = 13x + 10y$$

Question 2

A closed rectangular box has a volume of $$972\ \mathrm{cm}^3$$ and a width three times its length. Which set of dimensions minimizes the surface area?

Accepted Answer

$$6\ \mathrm{cm} 	imes 18\ \mathrm{cm} 	imes 9\ \mathrm{cm}$$

Question 3

An open-top box is made by cutting a square of side $$x$$ from each corner of an $$18'' 	imes 18''$$ sheet and folding up the sides. What value of $$x$$ maximizes the volume of the box?

Accepted Answer

$$x = 3$$ inches

Question 4

You are 1 mile north of a river that runs east-west. Your cabin is 8 miles east and 2 miles north of your current position. If you must stop at the river for water, which function represents the total distance $$D$$ you must walk as a function of $$x$$, the number of miles east you travel before reaching the river?

Accepted Answer

$$D = \sqrt{x^2$ + 1} + \sqrt{(8-x)^2$ + 9}$$

Question 5

A spherical balloon is being inflated so that its volume increases at $$24$$ cubic inches per second. When the radius is $$4$$ inches, what is the rate of change of the radius?

Accepted Answer

$$0.119 in/sec$$

Question 6

A 25-ft ladder is sliding down a wall. When the top is 15 ft above the ground and the bottom is moving away from the wall at $$2 ft/sec$$, how fast is the top of the ladder descending at that instant?

Accepted Answer

$$-2.667 ft/sec$$

Question 7

A helicopter flies away from a base at $$100$$ mph at a constant altitude of $$3$$ miles. When it is $$4$$ miles horizontally from the base, how fast is the direct distance from the helicopter to the base increasing?

Accepted Answer

$$80$$ mph

Question 8

A leaky ceiling drips water onto the floor, forming a circular puddle whose area increases at $$6$$ square inches per minute. When the radius is $$8$$ inches, how fast is the radius increasing?

Accepted Answer

$$0.119 in/$$min

Question 9

A conical reservoir has a diameter four times its height. Water is draining at $$8 m$^3/$$min when the water is $$5 m $$deep. At what rate is the water level dropping at that instant?

Accepted Answer

$$-0.0255 m/$$min

Question 10

Two sticks of lengths $$10 cm $$and $$12 cm $$form a triangle with the ground. When the angle between them is $$0.8$$ radians and increasing at $$0.1$$ radians/sec, how fast is the area of the triangle increasing?

Accepted Answer

$$4.180 cm$^2/sec$$

Question 11

A man $$6 ft $$tall walks toward a $$18 ft $$light post at $$5 ft/sec. $$How fast is the tip of his shadow approaching the light post?

Accepted Answer

$$7.5 ft/sec$$

Question 12

Find the two positive numbers whose product is $$200$$ and whose sum is minimized.

Accepted Answer

$$x = y = \sqrt{200}$$

Question 13

Find the two positive numbers whose product is $$200$$ and whose sum $$S = x + \frac{1}{2}y is $$minimized.

Accepted Answer

$$x = 10, y = 20$$

Question 14

A hydraulic press flattens an aluminum cylinder with constant volume $$75\pi cm$^3. If $$the height decreases at $$2 cm/sec$$, what is the rate of change of the radius when $$r = 5 cm $$and $$h = 3 cm$$?

Accepted Answer

$$\frac{5}{3} cm/sec$$

Question 15

Find the point on the curve $$f(x) = \sqrt{x}$$ that is closest to the point $$(3, 0). $$Which $$x-$$coordinate gives the minimum distance?

Accepted Answer

$$x = 2.5$$

Question 16

What is the maximal area of a rectangle inscribed between the $$x-$$axis and the curve $$f(x) = \sqrt{50 - x^2$}$$?

Accepted Answer

$$50$$ square units

Question 17

A 25-ft ladder leans against a wall. If $$x is $$the distance from the wall to the bottom of the ladder and $$y is $$the height of the top above the ground, which equation relates $$x$$, $$y$$, and the ladder's length?

Accepted Answer

$$x^2$$ + $$y^2$$ = $$25^2$$

Question 18

A man walks toward a light post, casting a shadow. If $$s is $$the distance from the tip of his shadow to the light post, which similarity ratio relates the lengths involved?

Accepted Answer

$$\frac{s}{18} = \frac{x}{6} ($$where $$x is $$the distance from the man to the light post)

Calculus I: Optimization and Related Rates Study Guidance

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