1. MTH1170 Related Rates

2. Preliminary
In differential calculus, related rates problems involve finding the rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The rates of change are always with respect to time.

3. The Procedure
1. Identify one or more equations linking the variable quantities.
2. Differentiate both sides of the equation (implicitly) with respect to time. Regard all variable quantities as a function of time.
3. Substitute the known rates of change and the known quantities into the equation.
4. Solve for the desired rate of change by substituting in the known variables.
It’s important to know that all of the variable in the formula are changing with time. This means that the given variables are only valid for one instance in time. A entire new set of values will be needed if you want to solve for a rate of change at a different point in time.

4. Example
How fast is the area of a rectangle changing if one side is 10cm long and increasing at a rate of 2cm/s, and the other side is 8 cm long and changing at -3cm/s.

5. Example – Oil Spil
The radius of an oil spil is increasing at 1m/s. How fast is the area increasing when r = 30m?
First we need to find a formula that relates the variable quantities. We need a formula that relates the area of a circle to its radius.

6. Example
Next we differentiate the function implicitly with respect to time.

7. Example
Where the variables represent the following:

8. Example
We do not need to rearrange the formula because it is already solved for the value we are looking for.
Next we substitute the given values into our formula and solve.

9. Example
A 10-meter ladder is leaning against the wall of a building, and the base of the ladder is sliding away from the building at a rate of 3 meters per second. How fast is the top of the ladder sliding down the wall when the base of the ladder is 6 meters from the wall?