Question

A solid sphere is melted and recast into a right circular cone with a base radius equal to the radius of the sphere. What is the ratio of the height and radius of the cone so formed?

• A. 4:3
• B. 2:3
• C. 3:4
• D. None of these

Answer 1

The Correct Option is D

To find the ratio of the height and radius of the cone formed after the solid sphere is melted and recast, we begin by equating the volumes of the original sphere and the new cone. 

The volume \( V \) of a sphere is given by:

\( V = \frac{4}{3}\pi r^3 \)

where \( r \) is the radius of the sphere.

The volume \( V \) of a cone is given by:

\( V = \frac{1}{3}\pi r^2 h \)

where \( r \) is the radius of the cone's base (equal to the sphere's radius) and \( h \) is the height of the cone.

Since the volume of the sphere equals the volume of the cone:

\(\frac{4}{3}\pi r^3 = \frac{1}{3}\pi r^2 h \)

To find \( h \), cancel \(\pi\) and one \( r^2 \) from both sides:

\(4r = h\)

Thus, \( h = 4r \).

The ratio of the height \( h \) and the radius \( r \) of the cone is:

(\( h:r \)) = 4:1.

Therefore, the correct answer is "None of these" as this ratio is not listed in the options provided.