The slant height of a conical mountain is 2.5 km and the area of its base is `1.54 km^2` . Taking ` pi= 22/7`, the height of the mountain is:

Options

A. 2.2 km
B. 2.4km
C. 3 km
D. 3.11 km

Answer

To find the height of the conical mountain given the slant height and the area of its base, we can follow these steps:

### Step 1: Understand the given information
- Slant height (l) = 2.5 km
- Area of the base (A) = 1.54 km²
- We will use the formula for the area of a circle to find the radius (r) of the base.

### Step 2: Use the area formula to find the radius
The area of the base of the cone is given by the formula:
\[ A = \pi r^2 \]

Substituting the values we have:
\[ 1.54 = \frac{22}{7} r^2 \]

### Step 3: Solve for \( r^2 \)
To isolate \( r^2 \), we can multiply both sides by \( \frac{7}{22} \):
\[ r^2 = 1.54 \times \frac{7}{22} \]

Calculating the right side:
\[ r^2 = \frac{1.54 \times 7}{22} = \frac{10.78}{22} = 0.49 \]

### Step 4: Calculate the radius \( r \)
Now, take the square root of \( r^2 \):
\[ r = \sqrt{0.49} = 0.7 \text{ km} \]

### Step 5: Use the Pythagorean theorem to find the height
In a right triangle formed by the height (h), radius (r), and slant height (l), we have:
\[ l^2 = r^2 + h^2 \]

Substituting the known values:
\[ (2.5)^2 = (0.7)^2 + h^2 \]

Calculating \( (2.5)^2 \) and \( (0.7)^2 \):
\[ 6.25 = 0.49 + h^2 \]

### Step 6: Solve for \( h^2 \)
Rearranging the equation gives:
\[ h^2 = 6.25 - 0.49 = 5.76 \]

### Step 7: Calculate the height \( h \)
Taking the square root of \( h^2 \):
\[ h = \sqrt{5.76} = 2.4 \text{ km} \]

### Final Answer
The height of the mountain is **2.4 km**.

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