Question

Wyatt saw a tower in the distance. When looking at the tower, his eye level makes a 72 degree angle with the top of the tower. Given that the tower is 96 meters tall and Wyatt's eye level is 1.8 meters from the ground, find the direct distance from Wyatt's eye level to the top of the tower. Round your answer to the nearest whole number. (1 point) which answer is correct and why

Answer 1

To find the direct distance from Wyatt's eye level to the top of the tower, we can use trigonometry.

1. **Determine the height difference**:
The height of the tower is 96 meters, and Wyatt's eye level is 1.8 meters. The height difference (h) between the top of the tower and Wyatt's eye level is:
\[
h = 96\, \text{m} - 1.8\, \text{m} = 94.2\, \text{m}
\]

2. **Use the tangent function**:
Wyatt makes a 72-degree angle with the top of the tower. We can use the tangent function for this right triangle formed by Wyatt's line of sight:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Here, the opposite side is the height difference (94.2 m), and the adjacent side (d) is the horizontal distance from Wyatt to the base of the tower.

Hence,
\[
\tan(72^\circ) = \frac{94.2}{d}
\]
Rearranging this gives:
\[
d = \frac{94.2}{\tan(72^\circ)}
\]

3. **Calculate the adjacent side (d)**:
Using a calculator for \(\tan(72^\circ) \approx 3.0777\),
\[
d \approx \frac{94.2}{3.0777} \approx 30.6\, \text{meters}
\]

4. **Find the direct distance (hypotenuse)**:
To find the direct distance (let's call it D) from Wyatt's eye level to the top of the tower, we can use the Pythagorean theorem:
\[
D = \sqrt{(94.2)^2 + (d)^2}
\]
Substituting \(d \approx 30.6\):
\[
D \approx \sqrt{(94.2)^2 + (30.6)^2}
\]

Calculating this gives:
\[
D \approx \sqrt{(94.2)^2 + (30.6)^2} \approx \sqrt{8868.64 + 936.36} \approx \sqrt{9805} \approx 99.02
\]

5. **Round to the nearest whole number**:
Therefore, the direct distance from Wyatt's eye level to the top of the tower is approximately 99 meters when rounded to the nearest whole number.

Thus, the final answer is **99 meters**.