1. RATES OF CHANGE:
GEOMETRIC

2. Rates of Change
Take the derivative with respect to whatever the variable is in the equation.
Ex) Find the rate of change of area of a circle with respect to radius when the radius is 1 cm.
A: Area
r: Radius

3. Related Rates
A word problem that are solved by implicit differentiation, relation is always in respect to time.
They have to do with changes in two or more variables with respect to time.
The variables are related such that when one changes the other must also change.

4. Step One
Ex) You have a spherical snowball and allow it to melt. Changes occur both to its radius and volume as it melts. We know that the volume is related to the radius by
Since volume and radius change over time, take the derivative of the volume formula with respect to time.
Rate of change of volume Rate of change of radius Derivative
with respect to time with respect to time equals

5. Step Two
2 takes the place of r and 1.5cm\min takes the place of
Now say we want to find the rate of change of volume when the radius is 2cm and the radius is changing at 1.5cm\minute.
2. Identify where the given values should go within the related rates

6. Step Three
3. Use algebra to solve for missing variable
Answer is:

7. Example One!
Find the rate of change of volume, with respect to side length of a cube when the side length is 8cm.

8. Example 2
Water is flowing out of a cylindrical storage tank at a rate of
If the tank has a radius of 8m, how fast is the water level falling?
because the radius
doesn’t change

9. A water tank is in the shape of a cone with a height of 5m and a diameter of 6 m at the top. Water is being pumped into the tank at a rate of Find the rate at which the water level is rising when the water is two meters deep.
Example 3

10. SUMMARY
Related rates are word problems that are solved by implicit differentiation. The relation is always in respect to time.
The first step in a related rates problem is to take the derivative.
Next, identify where the given values should go with the equation.
Lastly, remember to use algebra to solve for the missing variable.
Keep track of what equations to use. Try not to get area confused with volume. Remember to use correct units in your answer.
Hints: draw pictures and label them; write down
equations that relate the variables in the problem; DIFFERENTIATE; substitute in the information and solve.