# Question #a685f

##### 1 Answer

The time at which the sphere is increasing at

is at

The radius is

#### Explanation:

I assume by the "plane" surface area you are referring to the flat surface of the hemisphere. The flat surface will clearly be a circle whose area is given by:

In the above we have made it explicit that

We have that

Integrate this with respect to

Assuming (as no other initial condition is stated) that the radius of the sphere is

So

Using this information we can calculate the time derivative of

So, to work out the radius at which the circle's area is increasing at

So the time at which the sphere is increasing at

is at

We know from above that

So the radius is