1: Identify and write down all the key information:
Let r=radius of the circle, A=area of the circle and t=time. We have been given dr/dt
(the rate in which the radius is changing with respect to time) and the question wants us to find dA/dt
(the rate in which the area is changing with respect to time) when r=10.
2: Set up your chain rule equation:
We want our resulting rate to be dA/dt so we can place that on one side of the equation. We are given dr/dt so we should put this on the other side of the equation. From previous sections you should be able to complete the chain rule equation and instantly recognize that if we multiply dr/dt by dA/dr we will get our desired dA/dt.
3: Using given information in the question, work out the unknown rate:
We have just introduced dA/dr into the question, but how do we find out what it is? We need to find out an equation for A in terms of r which we can differentiate. The key information is in the question. The question says “radius of a circular puddle of water”
This means that:
4: Bring all the information back into the chain rule equation:
Now that we have found what dA/dr is, we can substitute that into the equation we made in Step 3.
This means that our rate of change of the area is dependent on the radius of the circle. When the radius of the circle is smaller, the dA/dt is smaller and when the radius of the circle is larger, the dA/dt is larger.
5: Substitute in any given conditions:
We have found dA/dt but that isn’t our final answer. The question asked for dA/dt when r=10, so we must substitute r=10 into dA/dt.
Hence when the radius of the puddle is 10cm, the rate of change of the area is
.