Question

Rashaad is visiting the Big Ben clock tower in London. When observing 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 Rashaad 's eye level is 1.8 meters from the ground, apply the sine ratio to find the direct distance from Rashaad's eye level to the top of the tower. Round your answer to the nearest whole number.

Answer 1

To find the direct distance from Rashaad's eye level to the top of the Big Ben clock tower, we can use the sine ratio from trigonometry.

1. **Determine the height of the tower above Rashaad's eye level**:
The total height of the tower is 96 meters, and Rashaad's eye level is 1.8 meters from the ground. Therefore, the height of the tower above Rashaad's eye level is:
\[
\text{Height above eye level} = 96 \, \text{meters} - 1.8 \, \text{meters} = 94.2 \, \text{meters}
\]

2. **Use the sine ratio**:
We have the opposite side (the height above eye level) and need to find the hypotenuse (the direct distance to the top of the tower).

The sine ratio is given by:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Where:
- \(\theta = 72^\circ\)
- Opposite = 94.2 meters
- Hypotenuse = x (the distance we want to find)

Plugging in the values, we have:
\[
\sin(72^\circ) = \frac{94.2}{x}
\]

3. **Rearranging to find x**:
\[
x = \frac{94.2}{\sin(72^\circ)}
\]

4. **Calculating \(\sin(72^\circ)\)**:
Using a calculator, we find:
\[
\sin(72^\circ) \approx 0.9511
\]

5. **Calculating x**:
\[
x = \frac{94.2}{0.9511} \approx 99.02
\]

6. **Round to the nearest whole number**:
\[
x \approx 99
\]

Therefore, the direct distance from Rashaad's eye level to the top of the Big Ben clock tower is approximately **99 meters**.