From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest `cm^2` .

Options

A. 15 cm^2
B. 18 cm^2
C. 19 cm^2
D. 13 cm^2

Answer

To find the total surface area of the remaining solid after hollowing out a conical cavity from a solid cylinder, we will follow these steps:

### Step 1: Determine the dimensions of the cylinder and cone
- **Height of the cylinder (H)** = 2.4 cm
- **Diameter of the cylinder** = 1.4 cm, hence **Radius (R)** = Diameter/2 = 1.4 cm / 2 = 0.7 cm

### Step 2: Calculate the curved surface area (CSA) of the cylinder
The formula for the curved surface area of a cylinder is:
\[ \text{CSA of Cylinder} = 2 \pi R H \]
Substituting the values:
\[ \text{CSA of Cylinder} = 2 \times \frac{22}{7} \times 0.7 \times 2.4 \]
Calculating:
\[ = 2 \times \frac{22}{7} \times 0.7 \times 2.4 = 10.56 \, \text{cm}^2 \]

### Step 3: Calculate the area of the base of the cylinder
The area of the base of the cylinder is given by:
\[ \text{Area of Base} = \pi R^2 \]
Substituting the values:
\[ \text{Area of Base} = \frac{22}{7} \times (0.7)^2 \]
Calculating:
\[ = \frac{22}{7} \times 0.49 = 1.54 \, \text{cm}^2 \]

### Step 4: Calculate the slant height (L) of the cone
Using the Pythagorean theorem, the slant height \(L\) of the cone can be calculated as:
\[ L = \sqrt{H^2 + R^2} \]
Substituting the values:
\[ L = \sqrt{(2.4)^2 + (0.7)^2} = \sqrt{5.76 + 0.49} = \sqrt{6.25} = 2.5 \, \text{cm} \]

### Step 5: Calculate the curved surface area (CSA) of the cone
The formula for the curved surface area of the cone is:
\[ \text{CSA of Cone} = \pi R L \]
Substituting the values:
\[ \text{CSA of Cone} = \frac{22}{7} \times 0.7 \times 2.5 \]
Calculating:
\[ = \frac{22}{7} \times 0.7 \times 2.5 = 5.5 \, \text{cm}^2 \]

### Step 6: Calculate the total surface area of the remaining solid
The total surface area of the remaining solid is given by:
\[ \text{Total Surface Area} = \text{CSA of Cylinder} + \text{Area of Base} + \text{CSA of Cone} \]
Substituting the values:
\[ \text{Total Surface Area} = 10.56 + 1.54 + 5.5 \]
Calculating:
\[ = 17.6 \, \text{cm}^2 \]

### Step 7: Round off the total surface area
Rounding off to the nearest cm²:
\[ \text{Total Surface Area} \approx 18 \, \text{cm}^2 \]

### Final Answer
The total surface area of the remaining solid is approximately **18 cm²**.