A cylinder is inscribed in a right circular cone of height 5.5 and radius (at the base) equal to 7. What are the dimensions of such a cylinder which has maximum volume?

Center a cylinder of radius r on the axis of the cone of height H and base radius R. If the cylinder has height h, using similar triangles,

H/R = (H-h)/r
h = H - Hr/R

volume of cylinder is

v = πr2h = πr2(H - Hr/R)
= πHr2 - πH/R r3

dv/dr = 2πHr - 3πH/R r2
= πHr(2 - 3r/R)

max at r = 2R/3, h = H/3

volume = πr2h = π(2R/3)2(H/3) = 4πR2H/27

To find the dimensions of the cylinder with maximum volume inscribed in a right circular cone, you can follow these steps:

Step 1: Understand the problem and visualize
Make sure you understand the problem and visualize the given cone. The cone has a height of 5.5 and a radius (at the base) equal to 7.

Step 2: Determine the dimensions of the cylinder
To find the dimensions of the cylinder, we need to consider the relationship between the cone and cylinder. When a cylinder is inscribed in a cone, the height of the cylinder is equal to the height of the cone, and the radius of the cylinder is equal to the radius of the base of the cone.

So, in this case, the height of the cylinder will be 5.5, and the radius of the cylinder will be 7.

Step 3: Calculate the maximum volume of the cylinder
The volume of a cylinder can be calculated using the formula V = πr^2h, where r is the radius and h is the height.

For our cylinder, the radius (r) is 7 and the height (h) is 5.5. Now, we can calculate the maximum volume.

V = π(7)^2(5.5)
V = 382.5π

Therefore, the maximum volume of the cylinder inscribed in the given cone is 382.5π cubic units.

So, the dimensions of the cylinder with maximum volume are a height of 5.5 units and a radius of 7 units.