A cone of height 15 cm and base diameter 30 cm is carved out of a wooden sphere of radius 15 cm. The percentage of wasted wood is :

Options

A. `75%`
B. `50%`
C. `40%`
D. `25%`

Answer

To find the percentage of wasted wood when a cone is carved out of a wooden sphere, we need to calculate the volumes of both the sphere and the cone, and then determine the volume of wood that is wasted.

### Step-by-Step Solution:

1. **Calculate the Volume of the Sphere:**
The formula for the volume of a sphere is given by:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
Given the radius \( r = 15 \) cm, we can substitute this value into the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi (15)^3 \]
Calculating \( (15)^3 \):
\[ (15)^3 = 3375 \]
Now substituting back:
\[ V_{\text{sphere}} = \frac{4}{3} \pi (3375) = 4500 \pi \text{ cm}^3 \]

2. **Calculate the Volume of the Cone:**
The formula for the volume of a cone is given by:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
The diameter of the cone is 30 cm, so the radius \( r \) is:
\[ r = \frac{30}{2} = 15 \text{ cm} \]
The height \( h \) of the cone is given as 15 cm. Now substituting these values into the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi (15)^2 (15) \]
Calculating \( (15)^2 \):
\[ (15)^2 = 225 \]
Now substituting back:
\[ V_{\text{cone}} = \frac{1}{3} \pi (225)(15) = \frac{1}{3} \pi (3375) = 1125 \pi \text{ cm}^3 \]

3. **Calculate the Volume of Wasted Wood:**
The volume of wasted wood is the volume of the sphere minus the volume of the cone:
\[ V_{\text{waste}} = V_{\text{sphere}} - V_{\text{cone}} = 4500 \pi - 1125 \pi = (4500 - 1125) \pi = 3375 \pi \text{ cm}^3 \]

4. **Calculate the Percentage of Wasted Wood:**
The percentage of wasted wood can be calculated using the formula:
\[ \text{Percentage of waste} = \left( \frac{V_{\text{waste}}}{V_{\text{sphere}}} \right) \times 100 \]
Substituting the values we calculated:
\[ \text{Percentage of waste} = \left( \frac{3375 \pi}{4500 \pi} \right) \times 100 \]
The \( \pi \) cancels out:
\[ \text{Percentage of waste} = \left( \frac{3375}{4500} \right) \times 100 = \frac{3375 \times 100}{4500} \]
Simplifying the fraction:
\[ = \frac{3375}{4500} = \frac{75}{100} = 0.75 \times 100 = 75\% \]

Thus, the percentage of wasted wood is **75%**.