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)
meters
meters
Answers
GPT-4o mini
To find the direct distance from Wyatt's eye level to the top of the tower, we can use trigonometry. We know the following:
- The height of the tower is 96 meters.
- Wyatt's eye level is 1.8 meters above the ground.
- The angle of elevation from Wyatt's eye level to the top of the tower is 72 degrees.
First, we need to calculate the height difference between Wyatt's eye level and the top of the tower:
\[
\text{Height difference} = \text{Height of the tower} - \text{Wyatt's eye level} = 96 \text{ m} - 1.8 \text{ m} = 94.2 \text{ m}
\]
Next, we will denote the direct distance (hypotenuse) from Wyatt's eye level to the top of the tower as \( d \). We can relate this to the height difference using the sine of the angle of elevation:
\[
\sin(72^\circ) = \frac{\text{Height difference}}{d}
\]
Rearranging the equation gives:
\[
d = \frac{\text{Height difference}}{\sin(72^\circ)}
\]
Substituting in the height difference we calculated:
\[
d = \frac{94.2}{\sin(72^\circ)}
\]
Now, we need to find \(\sin(72^\circ)\). Using a calculator, we find:
\[
\sin(72^\circ) \approx 0.9511
\]
Now substituting this value into the equation for \( d \):
\[
d = \frac{94.2}{0.9511} \approx 99.03
\]
Finally, rounding to the nearest whole number, we find:
\[
d \approx 99 \text{ meters}
\]
Thus, the direct distance from Wyatt's eye level to the top of the tower is approximately **99 meters**.
- The height of the tower is 96 meters.
- Wyatt's eye level is 1.8 meters above the ground.
- The angle of elevation from Wyatt's eye level to the top of the tower is 72 degrees.
First, we need to calculate the height difference between Wyatt's eye level and the top of the tower:
\[
\text{Height difference} = \text{Height of the tower} - \text{Wyatt's eye level} = 96 \text{ m} - 1.8 \text{ m} = 94.2 \text{ m}
\]
Next, we will denote the direct distance (hypotenuse) from Wyatt's eye level to the top of the tower as \( d \). We can relate this to the height difference using the sine of the angle of elevation:
\[
\sin(72^\circ) = \frac{\text{Height difference}}{d}
\]
Rearranging the equation gives:
\[
d = \frac{\text{Height difference}}{\sin(72^\circ)}
\]
Substituting in the height difference we calculated:
\[
d = \frac{94.2}{\sin(72^\circ)}
\]
Now, we need to find \(\sin(72^\circ)\). Using a calculator, we find:
\[
\sin(72^\circ) \approx 0.9511
\]
Now substituting this value into the equation for \( d \):
\[
d = \frac{94.2}{0.9511} \approx 99.03
\]
Finally, rounding to the nearest whole number, we find:
\[
d \approx 99 \text{ meters}
\]
Thus, the direct distance from Wyatt's eye level to the top of the tower is approximately **99 meters**.