A right circular cylinder is inscribed in a sphere of radius . Find the largest possible surface area of such a cylinder. Round the result to the nearest hundredth.

Question

A right circular cylinder is inscribed in a sphere of radius  . Find the largest possible surface area of such a cylinder. Round the result to the nearest hundredth.

Quizplus · Accepted Answer

Let the radius and height of the cylinder be $r$ and $h$, respectively. Since the cylinder is inscribed in the sphere, the diameter of the sphere is equal to the diagonal of the cylinder, so $2r \sqrt{2}=22eaa3e5_55bc_8ded_9f8f_379fa5d682dd_TB5971_11$ which implies $r=11eaa3e5_55bc_8ded_9f8f_379fa5d682dd_TB5971_11\sqrt{2}$. 
Now, the surface area of the cylinder is $2\pi r^2 + 2 \pi rh$, so we want to maximize $2\pi r^2 + 2 \pi rh = 22\pi r^2$. Substituting for $r$, we get $22\pi (11eaa3e5_55bc_8ded_9f8f_379fa5d682dd_TB5971_11\sqrt{2})^2 \approx 3063.6$. Rounded to the nearest hundredth, this is $3063.60$, which matches choice (D).

A Right Circular Cylinder Is Inscribed in a Sphere of Radius