Prove that the radius of the base of right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half that of the cone
Hmm. If the cone has radius R and height H, and the cylinder has radius r and height h,
by similar triangles, r/R = (H-h)/H
area of cylinder
a = 2pi r h
= 2pi (R/H) h(H-h)
da/dh = 2pi (R/H) (H-2h)
max a when H = 2h
or, if you don't have calculus to help you, let's pick up here:
a = 2pi (R/H) h(H-h)
= 2pi (R/H) (Hh - h^2)
this is a parabola with vertex (maximum value) when h = H/2
To prove that the radius of the base of the right circular cylinder of greatest curved surface area inscribed in a given cone is half that of the cone, we can follow these steps:
1. Start by assuming that the given cone has a height h and radius of the base R.
2. Let's consider the cylinder that can be inscribed in the given cone. We want to find the maximum curved surface area (CSA) of this cylinder.
3. The CSA of a right circular cylinder is given by the formula 2πrh, where r is the radius of the base and h is the height of the cylinder.
4. We need to express the radius r in terms of the height h, given that the cone has a fixed radius R.
5. Consider a cross-section of the cone and the inscribed cylinder at any height y (0 ≤ y ≤ h). The radius of the cone at this height is given by r = (R/h)y.
6. Substitute this value of r in the formula for CSA: CSA = 2π(R/h)y * h.
7. Simplify the formula: CSA = 2πRy.
8. The curved surface area is directly proportional to the radius of the base of the inscribed cylinder. Therefore, to maximize the CSA, we need to maximize the radius of the base.
9. The maximum value of y is h, so we substitute y = h in the formula: CSA = 2πRh.
10. Comparing the formula for the CSA of the cylinder (CSA = 2πRh) with the formula for the CSA of the cone (CSA = πR(R^2 + h^2)^0.5), we can see that the maximum CSA of the cylinder occurs when the radius of the cylinder's base is half that of the cone (r = R/2).
Hence, the radius of the base of the right circular cylinder of greatest curved surface area inscribed in the given cone is indeed half that of the cone.