A cone of height 2.8 cm has a lateral surface area `23.10 cm^2`. The radius of the base is :

Options

A. 3.5 cm
B. 2 cm
C. 2.1 cm
D. 4 cm

Answer

To find the radius of the base of the cone given its height and lateral surface area, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values:**
- Height (H) of the cone = 2.8 cm
- Lateral Surface Area (LSA) = 23.10 cm²

2. **Recall the formula for the lateral surface area of a cone:**
\[ \text{LSA} = \pi R L \]
where \( R \) is the radius of the base and \( L \) is the slant height.

3. **Express the slant height (L) in terms of radius (R):**
The slant height can be calculated using the Pythagorean theorem:
\[ L = \sqrt{H^2 + R^2} \]
Substituting the height:
\[ L = \sqrt{(2.8)^2 + R^2} = \sqrt{7.84 + R^2} \]

4. **Substitute L into the LSA formula:**
\[ 23.10 = \pi R \sqrt{7.84 + R^2} \]
Using \( \pi \approx 3.14 \):
\[ 23.10 = 3.14 R \sqrt{7.84 + R^2} \]

5. **Rearranging the equation:**
Divide both sides by \( 3.14 \):
\[ \frac{23.10}{3.14} = R \sqrt{7.84 + R^2} \]
Calculate \( \frac{23.10}{3.14} \):
\[ 7.35 = R \sqrt{7.84 + R^2} \]

6. **Square both sides to eliminate the square root:**
\[ (7.35)^2 = R^2 (7.84 + R^2) \]
Calculate \( (7.35)^2 \):
\[ 54.0225 = R^2 (7.84 + R^2) \]

7. **Expand and rearrange the equation:**
\[ 54.0225 = 7.84R^2 + R^4 \]
Rearranging gives:
\[ R^4 + 7.84R^2 - 54.0225 = 0 \]

8. **Let \( x = R^2 \):**
This transforms the equation into a quadratic:
\[ x^2 + 7.84x - 54.0225 = 0 \]

9. **Use the quadratic formula to solve for \( x \):**
The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 7.84, c = -54.0225 \):
\[ x = \frac{-7.84 \pm \sqrt{(7.84)^2 - 4 \cdot 1 \cdot (-54.0225)}}{2 \cdot 1} \]
Calculate the discriminant:
\[ (7.84)^2 + 4 \cdot 54.0225 = 61.4656 + 216.09 = 277.5556 \]
Now calculate:
\[ x = \frac{-7.84 \pm \sqrt{277.5556}}{2} \]
\[ x = \frac{-7.84 \pm 16.66}{2} \]

10. **Calculate the two possible values for \( x \):**
- \( x = \frac{8.82}{2} = 4.41 \) (valid since \( R^2 \) must be positive)
- \( x = \frac{-24.48}{2} \) (not valid)

11. **Find \( R \):**
\[ R^2 = 4.41 \implies R = \sqrt{4.41} \approx 2.1 \text{ cm} \]

### Final Answer:
The radius of the base of the cone is approximately \( 2.1 \) cm.