The total surface area of a solid hemisphere of diameter 14 cm is :

Question

Accepted Answer

To determine the total surface area of a solid hemisphere with a diameter of 14 cm, we need to consider both the curved surface area and the base area of the hemisphere.

1. **Calculate the radius of the hemisphere:**
   - The diameter of the hemisphere is given as 14 cm.
   - Therefore, the radius $ r $ is half of the diameter: 
     $$
     r = \frac{14}{2} = 7 \text{ cm}
     $$

2. **Calculate the curved surface area of the hemisphere:**
   - The formula for the curved surface area of a hemisphere is:
     $$
     2\pi r^2
     $$
   - Substituting the radius:
     $$
     2\pi (7)^2 = 2\pi \times 49 = 98\pi \text{ cm}^2
     $$

3. **Calculate the base area of the hemisphere:**
   - The base of the hemisphere is a circle with area:
     $$
     \pi r^2
     $$
   - Substituting the radius:
     $$
     \pi (7)^2 = \pi \times 49 = 49\pi \text{ cm}^2
     $$

4. **Calculate the total surface area of the hemisphere:**
   - The total surface area is the sum of the curved surface area and the base area:
     $$
     98\pi + 49\pi = 147\pi \text{ cm}^2
     $$

5. **Substitute the value of $\pi$ (approximately 3.14):**
   $$
   147\pi \approx 147 \times 3.14 = 461.58 \text{ cm}^2
   $$

6. **Round to the nearest whole number:**
   - The total surface area is approximately $ 462 \text{ cm}^2 $.

Therefore, the correct answer is Option 3: $ 462 \text{ cm}^2 $.

**Explanation for Incorrect Options:**

- **Option 1: $348 cm^2$**
  - This value is significantly lower than the calculated total surface area. It does not account for the correct combination of the curved surface area and the base area.

- **Option 2: $442 cm^2$**
  - This value is also lower than the calculated total surface area. It might be a result of an incorrect calculation or omission of part of the surface area.

- **Option 4: $478 cm^2$**
  - This value is higher than the calculated total surface area. It likely results from an overestimation or incorrect application of the formula.

In conclusion, the correct total surface area of the solid hemisphere is $ 462 \text{ cm}^2 $, which corresponds to Option 3.