A right circular cone of radius 4 cm and slant height 5 cm is carved out from a cylindrical piece of wood of same radius and height 5 cm. The surface area of the remaining wood is :

Options

A. `84 pi `
B. `70 pi `
C. `76 pi `
D. `50 pi `

Answer

To find the surface area of the remaining wood after carving out a right circular cone from a cylindrical piece of wood, we will follow these steps:

### Step 1: Identify the dimensions of the cylinder and the cone.
- The radius of the cylinder (and the cone) is \( r = 4 \) cm.
- The height of the cylinder is \( h = 5 \) cm.
- The slant height of the cone is \( l = 5 \) cm.

### Step 2: Calculate the height of the cone.
To find the height of the cone, we can use the Pythagorean theorem since we have the radius and the slant height. The formula is:
\[ l^2 = r^2 + h_{cone}^2 \]
Substituting the known values:
\[ 5^2 = 4^2 + h_{cone}^2 \]
\[ 25 = 16 + h_{cone}^2 \]
\[ h_{cone}^2 = 25 - 16 = 9 \]
\[ h_{cone} = \sqrt{9} = 3 \text{ cm} \]

### Step 3: Calculate the curved surface area of the cylinder.
The formula for the curved surface area (CSA) of a cylinder is:
\[ CSA_{cylinder} = 2 \pi r h \]
Substituting the values:
\[ CSA_{cylinder} = 2 \pi (4) (5) = 40\pi \text{ cm}^2 \]

### Step 4: Calculate the curved surface area of the cone.
The formula for the curved surface area of a cone is:
\[ CSA_{cone} = \pi r l \]
Substituting the values:
\[ CSA_{cone} = \pi (4) (5) = 20\pi \text{ cm}^2 \]

### Step 5: Calculate the area of the base of the cone.
The area of the base of the cone is given by:
\[ Area_{base} = \pi r^2 \]
Substituting the values:
\[ Area_{base} = \pi (4^2) = 16\pi \text{ cm}^2 \]

### Step 6: Calculate the total surface area of the remaining wood.
The total surface area of the remaining wood is the sum of the curved surface area of the cylinder, the curved surface area of the cone, and the area of the base of the cone:
\[ Total \, Surface \, Area = CSA_{cylinder} + CSA_{cone} + Area_{base} \]
Substituting the values:
\[ Total \, Surface \, Area = 40\pi + 20\pi + 16\pi = 76\pi \text{ cm}^2 \]

### Final Answer:
The surface area of the remaining wood is \( 76\pi \text{ cm}^2 \).

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