We are given a pole of height 33 m. Two points are located on opposite sides of the pole such that the angles of elevation from these points to the top of the pole are $53^\circ$ and $67^\circ$. The two points and the base of the pole are on the same horizontal level. We are asked to find the distance between the two points, correct to three significant figures.

1. Problem Description

We are given a pole of height 33 m. Two points are located on opposite sides of the pole such that the angles of elevation from these points to the top of the pole are $53^\circ$ and $67^\circ$. The two points and the base of the pole are on the same horizontal level. We are asked to find the distance between the two points, correct to three significant figures.

2. Solution Steps

Let $h$ be the height of the pole, which is 33 m. Let $d_1$ be the distance from the base of the pole to the first point, and let $d_2$ be the distance from the base of the pole to the second point. Let $\theta_1$ be the angle of elevation from the first point to the top of the pole, which is $53^\circ$. Let $\theta_2$ be the angle of elevation from the second point to the top of the pole, which is $67^\circ$.

We can use the tangent function to relate the height of the pole to the distances $d_1$ and $d_2$.

$$\tan(\theta_1) = \frac{h}{d_1}$$

$$\tan(\theta_2) = \frac{h}{d_2}$$

Solving for $d_1$ and $d_2$, we have:

$$d_1 = \frac{h}{\tan(\theta_1)}$$

$$d_2 = \frac{h}{\tan(\theta_2)}$$

Substituting the given values, we have:

$$d_1 = \frac{33}{\tan(53^\circ)} \approx \frac{33}{1.327} \approx 24.868$$

$$d_2 = \frac{33}{\tan(67^\circ)} \approx \frac{33}{2.356} \approx 14.007$$

The distance between the two points is $d = d_1 + d_2$.

$$d = d_1 + d_2 = \frac{33}{\tan(53^\circ)} + \frac{33}{\tan(67^\circ)} \approx 24.868 + 14.007 \approx 38.875$$

Rounding to three significant figures, the distance between the two points is 38.9 m.

3. Final Answer

38.9 m