Final answer:
To find the maximum volume of a cylinder inscribed in a cone, we must determine the relationship between the cone's dimensions and the cylinder's dimensions, use the volume formula for a cylinder, and apply calculus to find the dimensions of the cylinder that yield the maximum volume.
Step-by-step explanation:
The problem involves finding the dimensions of a cylinder that has the maximum volume when inscribed in a cone with specific dimensions. To solve for the maximum volume of the cylinder, we need to use calculus to find the critical points of the volume function of the inscribed cylinder.
Step-by-step approach:
- First, we write down the formula for the volume of a cylinder, V = πr²h.
- Using similarity of triangles in the cone and the cylinder, we can relate the height of the cylinder to its radius. If the cone's height is 10 inches and its base radius is 5 inches, the ratio of height to radius is always 2:1, so h = 2r.
- Substitute the value of h from the previous step into the volume formula to get a volume function in terms of r only: V = πr²(2r) = 2πr³.
- To find the maximum volume, we take the derivative of V with respect to r, set it equal to zero, and solve for r.
- Once we find the radius that gives the maximum volume, we use the ratio to find the corresponding height of the cylinder.
This will give us the dimensions of the cylinder that yields the maximum volume when inscribed inside the given cone.