Final answer:
To find the length of the radius that maximizes the volume of a closed-top cylinder, we can equate the surface area to a given value and then differentiate the volume function with respect to the radius to find the maximum volume.
Step-by-step explanation:
To find the length of the radius that maximizes the volume of a closed-top cylinder, we need to consider the relationship between the volume and the surface area. Since we are given that the cylinder is made from 33 square inches of material, we can write the surface area as:
2πrh + πr^2 = 33
We can solve this equation for h and substitute it into the formula for the volume of a cylinder. Then, we can differentiate the volume function with respect to r, set it equal to zero, solve for r, and find its value that maximizes the volume. By doing this, we can find the length of the radius that maximizes the volume of the cylinder.