From a solid sphere of radius 15 cm, a right circular cylindrical hole of radius 9 cm whose axis passing through the centre is removed. The total surface area of the remaining solid is :

Options

A. a. `1188 pi cm^(2) `
B. b. `108 pi cm^(2) `
C. `1170 pi cm^(2) `
D. `144 pi cm^(2) `

Answer

To find the total surface area of the remaining solid after removing a cylindrical hole from a solid sphere, we can follow these steps:

### Step 1: Calculate the surface area of the sphere
The formula for the surface area of a sphere is given by:
\[ \text{Surface Area of Sphere} = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

Given that the radius of the sphere is 15 cm:
\[ \text{Surface Area of Sphere} = 4\pi (15)^2 = 4\pi (225) = 900\pi \text{ cm}^2 \]

### Step 2: Calculate the height of the cylindrical hole
The height \( h \) of the cylindrical hole can be calculated using the Pythagorean theorem. The radius of the sphere \( R \) is 15 cm and the radius of the cylindrical hole \( r \) is 9 cm. The height \( h \) can be calculated as follows:
\[ h = 2\sqrt{R^2 - r^2} \]
Substituting the values:
\[ h = 2\sqrt{15^2 - 9^2} = 2\sqrt{225 - 81} = 2\sqrt{144} = 2 \times 12 = 24 \text{ cm} \]

### Step 3: Calculate the lateral surface area of the cylindrical hole
The lateral surface area of a cylinder is given by:
\[ \text{Lateral Surface Area of Cylinder} = 2\pi r h \]
Substituting the values:
\[ \text{Lateral Surface Area of Cylinder} = 2\pi (9)(24) = 432\pi \text{ cm}^2 \]

### Step 4: Calculate the total surface area of the remaining solid
The total surface area of the remaining solid is calculated by subtracting the lateral surface area of the cylinder from the surface area of the sphere and adding the area of the two circular ends of the cylinder (which are not part of the sphere):
\[ \text{Total Surface Area} = \text{Surface Area of Sphere} - \text{Lateral Surface Area of Cylinder} + 2 \times \text{Area of Circular Ends} \]
The area of one circular end is:
\[ \text{Area of Circular End} = \pi r^2 = \pi (9)^2 = 81\pi \text{ cm}^2 \]
Thus, the area of two circular ends is:
\[ 2 \times 81\pi = 162\pi \text{ cm}^2 \]
Now substituting everything into the total surface area formula:
\[ \text{Total Surface Area} = 900\pi - 432\pi + 162\pi = (900 - 432 + 162)\pi = 630\pi \text{ cm}^2 \]

### Final Answer
\[ \text{Total Surface Area of the remaining solid} = 630\pi \text{ cm}^2 \]

---