Related Rates

One thing changes, so another must too.

12 problems · about 23 minutes · CHA-3.E

NameDate

Figure — use these to answer the problems

Volume against side length is cubic. The steeper the curve, the faster volume responds to a change in side.

y = 1x³ + 0x + 0

Point of tangency2

Figure 1

Straightforward practice. Get the method working first.

5 problems

dV/dt = 3s² · ds/dt. With s = 3 and ds/dt = 2, what is dV/dt?
Answer
2
Why must substitution wait until after differentiating?
- A substituted value becomes a constant, and constants have zero derivative.
- It makes the algebra messier.
dV/dt = 3s² · ds/dt. With s = 4 and ds/dt = 2, what is dV/dt?
Answer
dA/dt = 2πr · dr/dt. With r = 5 and dr/dt = 2, what is dA/dt divided by π?
Answer
A tank drains. Is dV/dt positive or negative? 1 positive, 2 negative.
Answer

The same ideas with more to keep track of.

3 problems

A square’s area is A = s². dA/dt = 2s · ds/dt. With s = 6 and ds/dt = 3, what is dA/dt?
Answer
Should a changing quantity be substituted before differentiating? 1 for yes, 0 for no.
Answer
dV/dt = 3s² · ds/dt. With s = 1 and ds/dt = 5, what is dV/dt?
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Put the related rates method in order.
Write 1 to 4 in the boxes to put these in order.
- Draw the situation and label what changes.
- Write an equation relating the quantities.
- Differentiate both sides with respect to time.
- Substitute the known values and solve.
dA/dt = 2s · ds/dt. With s = 10 and ds/dt = 1, what is dA/dt?
Answer
The Cube
dV/dt = 3s² · ds/dt. With s = 2 and ds/dt = 1, what is dV/dt?
Answer
The Circle
A circle’s area is A = πr². dA/dt = 2πr · dr/dt. With r = 3 and dr/dt = 1, what is dA/dt divided by π?
Answer