Question

A sphere's radius that starts at 6 inches starts to increase at a constant rate of 3 inches per minute. The formula for the area of a sphere is A = 4 πr². What is the appropriate function notation to define the area of the sphere in terms of the size of the radius.

Answer 1

The formula for the area of a sphere is given as A = 4πr², where 'A' is the area and 'r' is the radius. In this case, the radius 'r' is increasing at a constant rate, so it's a function of time.

We can represent the radius as a function of time 't' (in minutes) as follows: r(t) = 6 + 3t. This is because the sphere's radius starts at 6 inches and then increases at a constant rate of 3 inches per minute.

Then, we can substitute r(t) into the area formula to get the area as a function of time. This results in the following:

A(t) = 4π[6 + 3t]²

So, the appropriate function notation to define the area of the sphere in terms of the radius size is A(t) = 4π[6 + 3t]².