Questions: Look at this cone: If the height is doubled and the radius is tripled, then which of the following statements about its volume will be true? The new volume will be 18 times the old volume. The new volume will be 5 times the old volume. The new volume will be 54 times the old volume. The new volume will be 6 times the old volume. Submit

Solution Steps

The volume \( V \) of a cone is given by the formula: \[ V = \frac{1}{3} \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.

Given the original radius \( r = 36 \) mm and height \( h = 80 \) mm: \[ V_{\text{original}} = \frac{1}{3} \pi (36)^2 (80) \]

The height is doubled, so the new height \( h_{\text{new}} = 2 \times 80 = 160 \) mm. The radius is tripled, so the new radius \( r_{\text{new}} = 3 \times 36 = 108 \) mm.

Using the new dimensions: \[ V_{\text{new}} = \frac{1}{3} \pi (108)^2 (160) \]

To find the factor by which the volume changes: \[ \frac{V_{\text{new}}}{V_{\text{original}}} = \frac{\frac{1}{3} \pi (108)^2 (160)}{\frac{1}{3} \pi (36)^2 (80)} \]

\[ \frac{V_{\text{new}}}{V_{\text{original}}} = \frac{(108)^2 (160)}{(36)^2 (80)} \] \[ = \frac{(3 \times 36)^2 (2 \times 80)}{(36)^2 (80)} \] \[ = \frac{9 \times (36)^2 \times 2 \times 80}{(36)^2 \times 80} \] \[ = 9 \times 2 \] \[ = 18 \]

Final Answer