A circular patch of oil spreads on water…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A circular patch of oil spreads on water. Its area is square centimetres (cm) when the radius is centimetres (cm). Find the rate at which the area changes with the radius when cm. Now suppose the radius grows as centimetres at time seconds (s). Write the area as a function of and find the rate at which the area grows at s, in square centimetres per second.
Need a reminder? The key ideaShow
Product rule: . Quotient rule: . The derivative is a rate of change, in units of per unit of .
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Given. in cm, and later .
Plan. Differentiate the area with respect to , then with respect to .
Step 1. With the radius. . At cm: per cm.
Step 2. As a function of time. in cm.
Step 3. With the time. . At s: per s.
Check. At s the radius is cm, and , which agrees. The Class 12 chapter explains this chain of rates.
Answer: per cm, and per s.
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