Problem Set 6—Related Rates Problems

Purpose

This lesson reinforces your understanding of how to use the derivative to solve related rates problems.

Background

This exercise is mainly based on material in section 3.8 (“Related Rates”) of our textbook. We covered (or will cover) this material in classes between approximately October 15 and October 17.

One exercise (Problem 2), however, is a review of implicit differentiation and the meaning of derivatives rather than a related rates problem. See section 3.7 (“Implicit Differentiation”) and lectures between October 2 and October 10 for a review of this material.

Activity

Solve each of the following problems:

Problem 1

Section 3.8, exercise 12 (Find the rate at which a cube’s volume increases given the rate at which its surface area increases).

Problem 2

Imagine that you are swinging a weight on a 1-meter long string around your head. To describe the path the weight follows, use a coordinate system whose origin is at the top center of your head, with the X axis running out your right shoulder and the Y axis perpendicular to your face, as shown here in a view from above the top of your head:

In this coordinate system, and measuring distances in meters, the weight’s path is described by the equation x² + y² = 1.

Now suppose you let go of the string when the weight is at point ( √2/2, -√2/2 ). Give the equation for the line that the weight flies away along. Assume there is no gravity pulling the weight down as it moves.

Problem 3

Section 3.8, exercise 20 (Find the rate at which a plate’s area changes as its radius changes.)

Problem 4

Section 3.8, exercise 36 (Find the rate at which the angle of the line connecting a point to the origin is changing as the point moves along a parabola.)

Problem 5

Section 3.8, exercise 30 (Show that a growing raindrop’s radius increases at a constant rate).

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.