Question

If the radius of the base of a right circular cone is \(3r\) and its height is equal to the radius of the base, then its volume is :

• A. \(\frac{1}{3}\pi r^3\)
• B. \(\frac{2}{3}\pi r^3\)
• C. \(3\pi r^3\)
• D. \(9\pi r^3\)

Hint:

1. Identify the cone's actual radius (\(R\)) and height (\(H\)) based on the problem statement: \(R_{\text{cone}} = 3r\) \(H_{\text{cone}} = \text{radius of base} = 3r\) 2. Use the cone volume formula: \(V = \frac{1}{3}\pi R_{\text{cone}}^2 H_{\text{cone}}\). 3. Substitute: \(V = \frac{1}{3}\pi (3r)^2 (3r)\). 4. Calculate: \(V = \frac{1}{3}\pi (9r^2)(3r) = \frac{1}{3}\pi (27r^3) = 9\pi r^3\).

Answer 1

The Correct Option is D

Concept: The volume of a right circular cone is given by the formula \(V = \frac{1}{3}\pi R^2 H\), where \(R\) is the radius of the base of the cone and \(H\) is the height of the cone. Step 1: Identify the given dimensions of the cone Let the radius of the base of this specific cone be \(R_{\text{cone}}\) and its height be \(H_{\text{cone}}\).
Radius of the base, \(R_{\text{cone}} = 3r\).
Height of the cone, \(H_{\text{cone}}\), is "equal to the radius of the base." This means \(H_{\text{cone}} = R_{\text{cone}} = 3r\). Step 2: Substitute these dimensions into the volume formula The volume formula for a cone is \(V = \frac{1}{3}\pi R_{\text{cone}}^2 H_{\text{cone}}\). Substitute \(R_{\text{cone}} = 3r\) and \(H_{\text{cone}} = 3r\): \[ V = \frac{1}{3}\pi (3r)^2 (3r) \] Step 3: Simplify the expression First, calculate \((3r)^2\): \[ (3r)^2 = 3^2 \times r^2 = 9r^2 \] Now substitute this back into the volume equation: \[ V = \frac{1}{3}\pi (9r^2) (3r) \] Multiply the terms: \[ V = \frac{1}{3}\pi (9 \times 3 \times r^2 \times r) \] \[ V = \frac{1}{3}\pi (27r^3) \] Now, multiply by \(\frac{1}{3}\): \[ V = \frac{27}{3}\pi r^3 \] \[ V = 9\pi r^3 \] The volume of the cone is \(9\pi r^3\).