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 the hemisphere and the cone are equal, find the ratio of radius and height of the cone.

Hint:

For solids combining cone and hemisphere, equate the curved surface areas using their formulas: \[ \text{CSA of cone} = \pi r l, \quad \text{CSA of hemisphere} = 2\pi r^2. \] Always apply Pythagoras theorem to relate slant height \( l \), height \( h \), and radius \( r \).

Answer 1

Step 1: Let the radius and height of the cone be \( r \) and \( h \) respectively.
Since the cone is surmounted on a hemisphere of the same base radius, both share the same \( r \).
Step 2: Write the formulae for curved surface areas.
Curved surface area (CSA) of a cone: \[ \text{CSA}_{\text{cone}} = \pi r l \] Curved surface area (CSA) of a hemisphere: \[ \text{CSA}_{\text{hemisphere}} = 2\pi r^2 \]
Step 3: Given that both curved surfaces are equal.
\[ \pi r l = 2\pi r^2 \]
Step 4: Simplify the equation.
Cancel \( \pi r \) (since \( r \neq 0 \)): \[ l = 2r \] Step 5: Use Pythagoras theorem in the cone.
For a cone, \[ l^2 = r^2 + h^2 \] Substitute \( l = 2r \): \[ (2r)^2 = r^2 + h^2 \] \[ 4r^2 = r^2 + h^2 \] \[ h^2 = 3r^2 \] \[ h = \sqrt{3}r \] Step 6: Find the required ratio.
\[ \text{Ratio of radius to height} = r : h = r : \sqrt{3}r = 1 : \sqrt{3} \] Step 7: Final Answer.
\[ \boxed{r : h = 1 : \sqrt{3}} \]