Question

The diameter of a sphere is equal to the height of the cone of equal volume. If r and R are the radii of cone and sphere respectively, then r² =

• A. \(2R^2\)
• B. \(\frac{R^2}{2}\)
• C. \(4R^2\)
• D. \(R^2\)

Answer 1

The Correct Option is A

Let the radius of the sphere be \( R \), and the radius of the cone be \( r \), with their volumes being equal. The volume of a sphere is given by: \[ V_{\text{sphere}} = \frac{4}{3} \pi R^3 \] The volume of a cone is given by: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] where \( h \) is the height of the cone, and we are given that the diameter of the sphere is equal to the height of the cone, so \( h = 2R \). Now, equating the two volumes since the volumes are equal: \[ \frac{4}{3} \pi R^3 = \frac{1}{3} \pi r^2 (2R) \] Simplifying this: \[ \frac{4}{3} \pi R^3 = \frac{2}{3} \pi r^2 R \] Canceling out \( \frac{1}{3} \pi R \) from both sides: \[ 4R^2 = 2r^2 \] Dividing both sides by 2: \[ 2R^2 = r^2 \]

The correct answer is option (A): \(2R^2\)