Question

A solid is in the shape of a cone which is surmounted on a hemisphere of same base radius. If the curved surfaces of hemisphere and cone are equal, find the ratio of radius and height of the cone.

Hint:

To find the ratio of the radius and height of a cone surmounted by a hemisphere with equal curved surface areas, equate the curved surface areas and use the Pythagorean theorem for the slant height.

Answer 1

Let the radius of the base of the cone and the hemisphere be \( r \), and let the height of the cone be \( h \).
Step 1: The formula for the curved surface area (CSA) of a hemisphere is: \[ \text{CSA of hemisphere} = 2 \pi r^2. \] The formula for the curved surface area of a cone is: \[ \text{CSA of cone} = \pi r l, \] where \( l \) is the slant height of the cone.
Step 2: We are given that the curved surfaces of the hemisphere and the cone are equal, so: \[ 2 \pi r^2 = \pi r l. \] Canceling \( \pi r \) from both sides: \[ 2r = l. \]
Step 3: Now, we know that the slant height \( l \) of the cone is related to the radius and height of the cone by the Pythagorean theorem: \[ l = \sqrt{r^2 + h^2}. \] Substitute \( l = 2r \) into this equation: \[ 2r = \sqrt{r^2 + h^2}. \]
Step 4: Square both sides: \[ 4r^2 = r^2 + h^2. \]
Step 5: Simplify and solve for \( h^2 \): \[ 4r^2 - r^2 = h^2 \quad \Rightarrow \quad 3r^2 = h^2 \quad \Rightarrow \quad h = \sqrt{3}r. \]
Conclusion: The ratio of the radius to the height of the cone is \( \frac{r}{h} = \frac{1}{\sqrt{3}} \).