Question 1

A 26-foot ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 2 ft/sec, how fast is the bottom of the ladder moving away from the wall when the bottom is 10 feet from the wall?

Accepted Answer

$$1.25 ft/sec$$

Question 2

Given the relationship $$x^2$$ + $$y^2$$ = $$26^2$$ for a ladder problem, if $$\frac{dy}{dt} = -2 ft/sec $$and $$x = 10 ft$$, $$y = 24 ft$$, what is $$\frac{dx}{dt}$$?

Accepted Answer

$$1.25 ft/sec$$

Question 3

A westbound plane travels at 120 knots and a northbound plane at 160 knots. One-half hour after the northbound plane passes over the airport, how fast is the distance between the planes changing?

Accepted Answer

$$56$$ knots

Question 4

Suppose a point moves along the circle $$x^2$$ + $$y^2 = 25$. At the point (-3$$, 4)$$, if $$\frac{dx}{dt} = 0.4$$ units/sec, what is $$\frac{dy}{dt}$$?

Accepted Answer

$$-0.3$$ units/sec

Question 5

A company’s cost function is $$C(x) = 4x + 10,000. If $$production increases at $$500$$ units/week when $$x = 2,000$$, what is the rate of increase in cost?

Accepted Answer

$$1,000$$ per week

Question 6

A company’s revenue function is $$R(x) = 6x. If $$production increases at $$500$$ units/week when $$x = 2,000$$, what is the rate of increase in revenue?

Accepted Answer

$$3,000$$ per week

Question 7

A company’s profit function is $$P(x) = 2x - 10,000. If $$production increases at $$500$$ units/week when $$x = 2,000$$, what is the rate of increase in profit?

Accepted Answer

$$2,000$$ per week

Question 8

Which of the following is the correct first step in solving a related rates problem?

Accepted Answer

Sketch a figure if helpful.

Question 9

If $$x$$ and $$y$$ are both functions of time and are related by $$x^2$$ + $$y^2 = 25$, what is the result of differentiating both sides with respect to t$?

Accepted Answer

$$2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0$$

Question 10

A company’s cost, revenue, and profit functions are $$C(x) = 4x + 10,000$$, $$R(x) = 6x$$, and $$P(x) = 2x - 10,000. If$$ $$\frac{dx}{dt} = 500$$, what is $$\frac{dP}{dt}$$?

Accepted Answer

$$2,000$$

Question 11

In the ladder problem, why is $$\frac{dy}{dt}$$ negative?

Accepted Answer

Because the top of the ladder is moving down the wall.

Question 12

If a company’s profit is increasing at $$2,000$$ per week, and the cost is increasing at $$1,000$$ per week, what is the rate of increase in revenue?

Accepted Answer

$$3,000$$ per week

Question 13

Which of the following best describes a related rates problem?

Accepted Answer

A problem where two or more quantities are changing with respect to time and are related by an equation.

Question 14

A point moves along the circle $$x^2$$ + $$y^2 = 25$. If x = -3$, y = 4$, and $$\frac{dx}{dt} = 0.4$$, what is the value of $$\frac{dy}{dt}$$?

Accepted Answer

$$-0.3$$

Question 15

In the airplane problem, if the westbound plane is 60 nautical miles from the airport and the northbound plane is 80 nautical miles from the airport, what is the distance between the planes?

Accepted Answer

$$100$$ nautical miles

Related Rates- Chapter 3

Study Guide - Practice Questions

Study Guide - Flashcards