The slant height of a conical mountain is 2.5 km and the area of its base is 1.54 km 2 . Find the height of the mountain.

Answer

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

### Step 1: Identify the given values
- Slant height (l) = 2.5 km
- Area of the base (A) = 1.54 km²

### Step 2: Relate the area of the base to the radius
The area of the base of a cone is given by the formula:
\[ A = \pi r^2 \]
Where \( r \) is the radius of the base.

Substituting the given area:
\[ 1.54 = \pi r^2 \]

Using \( \pi \approx \frac{22}{7} \):
\[ 1.54 = \frac{22}{7} r^2 \]

### Step 3: Solve for the radius (r)
To find \( r^2 \), we rearrange the equation:
\[ r^2 = \frac{1.54 \times 7}{22} \]

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

Taking the square root to find \( r \):
\[ r = \sqrt{0.49} \]
\[ r = 0.7 \text{ km} \]

### Step 4: Apply the Pythagorean theorem
In a right triangle formed by the height (h), radius (r), and slant height (l), we can use the Pythagorean theorem:
\[ l^2 = h^2 + r^2 \]

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

Calculating the squares:
\[ 6.25 = h^2 + 0.49 \]

### Step 5: Solve for the height (h)
Rearranging the equation to isolate \( h^2 \):
\[ h^2 = 6.25 - 0.49 \]
\[ h^2 = 5.76 \]

Taking the square root to find \( h \):
\[ h = \sqrt{5.76} \]
\[ h = 2.4 \text{ km} \]

### Conclusion
The height of the conical mountain is **2.4 km**.

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