First, lets look at how they introduced t into the equation. We are told, "At time t = 0 the spheres have radii of 6 cm and 40 cm. The radius of the smaller sphere is increasing at a rate of 8 cm/sec, while the radius of the larger sphere is increasing at a rate of 2 cm/sec." That means that you can describe the radii of the circles in terms of t since the radius is a function of time. Let R be the radius of the eternal circle and r be the radius of the internal circle.
R = 40 cm(the starting radius) + 2 cm/sec (the rate of expansion) *t sec(the time) = 40 + 2t
r = 6 cm(the starting radius) + 8cm/sec (the rate of expansion) *t sec(the time) = 6 + 8t
Those values for R and r are substituted into the standard equations for the volume of a circle.
To find the maximum value for V, you need to differentiate that equation in order to find and test the critical numbers.
The only critical value of t > 0 is 2, and testing will confirm that V is maximized at t = 2 seconds. Hope that helps.