A solid cone of height 9 cm with diameter of its base 18 cm is cut out from a wooden solid sphere of radius 9 cm. The percentage of wood wasted is :

Options

A. `25%`
B. `30%`
C. `50%`
D. `75%`

Answer

To solve the problem of finding the percentage of wood wasted when a solid cone is cut from a wooden solid sphere, we will follow these steps:

### Step 1: Calculate the volume of the sphere.
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Here, the radius \( r \) of the sphere is 9 cm.

Substituting the value of \( r \):
\[ V = \frac{4}{3} \pi (9)^3 \]
Calculating \( 9^3 \):
\[ 9^3 = 729 \]
Now substituting back:
\[ V = \frac{4}{3} \pi (729) = \frac{2916}{3} \pi = 972 \pi \text{ cm}^3 \]

### Step 2: Calculate the volume of the cone.
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
The diameter of the base of the cone is 18 cm, so the radius \( r \) is:
\[ r = \frac{18}{2} = 9 \text{ cm} \]
The height \( h \) of the cone is given as 9 cm.

Substituting the values into the formula:
\[ V = \frac{1}{3} \pi (9)^2 (9) \]
Calculating \( 9^2 \):
\[ 9^2 = 81 \]
Now substituting back:
\[ V = \frac{1}{3} \pi (81)(9) = \frac{729}{3} \pi = 243 \pi \text{ cm}^3 \]

### Step 3: Calculate the volume of wood wasted.
The volume of wood wasted is the volume of the sphere minus the volume of the cone:
\[ \text{Volume of wood wasted} = V_{\text{sphere}} - V_{\text{cone}} = 972 \pi - 243 \pi = (972 - 243) \pi = 729 \pi \text{ cm}^3 \]

### Step 4: Calculate the percentage of wood wasted.
To find the percentage of wood wasted, we use the formula:
\[ \text{Percentage of wood wasted} = \left( \frac{\text{Volume of wood wasted}}{\text{Volume of sphere}} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage of wood wasted} = \left( \frac{729 \pi}{972 \pi} \right) \times 100 \]
The \( \pi \) cancels out:
\[ = \left( \frac{729}{972} \right) \times 100 \]
Calculating the fraction:
\[ \frac{729}{972} = \frac{3}{4} \]
Now substituting back:
\[ = \frac{3}{4} \times 100 = 75\% \]

### Final Answer:
The percentage of wood wasted is **75%**.