Question

What is the height of a mountain if the angle of elevation from a point 2000 meters away from the base is 45°?

• A. 1000 meters
• B. 2500 meters
• C. 3000 meters
• D. 2000 meters

Answer 1

Answer: (D)

2000 meters

Answer 2

The problem describes a right-angled triangle where:

• The height of the mountain is the side opposite to the angle of elevation.
• The distance from the point to the base of the mountain is the side adjacent to the angle of elevation.
• The angle of elevation is given as 45°.
• The distance from the base is 2000 meters.

We need to find the height of the mountain. The trigonometric ratio that relates the opposite side, adjacent side, and the angle is the tangent function:

$$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$$

In this case:

• $\theta = 45^\circ$
• Opposite side = Height of the mountain (let's call it $h$)
• Adjacent side = 2000 meters

Substitute these values into the formula:

$$\tan(45^\circ) = \frac{h}{2000}$$

We know that $\tan(45^\circ) = 1$.

$$1 = \frac{h}{2000}$$

Now, solve for $h$:

$$h = 2000 \times 1$$ $$h = 2000 \text{ meters}$$

Thus, the height of the mountain is 2000 meters.