Related Rates

2-D Shape Formulas and 3-D Shape Formulas will be very important for these.

Abstract

The typical related rates problem:

You're usually given some parameter which changes at some rate, and the value for that parameter.
Usually, you are to find some other variable with a rate of change when said parameter is at that value.

Find an equation which has both the parameter and the variable you are looking for
Differentiating both sides w/ respect to time will give the rate of change for both the parameter and variable
Since you know the rate of change of that parameter (given), plug it in, then solve for the rate of change of the variable

Dr. Gail Walker

Sphere

A spherical balloon is being filled at a rate of
Determine the rate of change of the radius when the volume is

Solution
Find: at the instant when
Given:

Find when

So, when

Therefore the radius is increasing at a rate of at the instant when

Ladder

A 5m ladder rests against the side of a building. The base of the ladder begins to slip at a constant rate of . How fast is the top of the ladder sliding down the wall when the base of the ladder is 2m from the way?

Solution
Let rep. the position of the base of the ladder from the wall (m)
Let ret. the position of the top of the ladder from the wall (m)
Let represent the length of the ladder (m)

Find: at the instant when
Given:
Constant:

Find when

Now, when

Therefore, the position of the top of the ladder is falling at a rate of at the instant when

Cone

A conical reservoir is pilling with water at a constant rate of . The reservoir is deep and has a maximum diameter of . Calculate the rate at which the depth of water is increasing when the depth is

Solution

Find: ad the instant when
Given:

Therefore, the depth of the water is increasing at a rate of at the instant when

TOP G PROFESSOR JINTAO DENG

Sphere

A spherical balloon is being inflated at a rate of 1 litre per minute. How quickly is the diameter of the balloon increasing at the moment when the diameter is 10cm?

We know and , so .
Differentiating both sides, we get
Now we just plug in and , and make sure the units match. Since 1 litre is a cubic decimetre, we can express the diameter as 1 dm. This gives . So

Airplane

An aircraft is cruising at an altitude of 10 km, at a groundspeed of 900 km/hr. An observer on the ground sees it pass directly overhead, and watches it travel away from him. When the angle of elevation reaches , how quickly is it decreasing?

Given:

An altitude of (const)
The horizontal component of the distance from the observer, is changing at the rate
The angle of elevation, , is radians

We are asked to find .
Since (see Trig Functions and Differentiation Rules), differentiating with respect to time gives:

When , we have , and so