To find the height from the top of the slide to the ground, we can use trigonometry. The angle of depression is given as 31.66°, and the length of the slide is 46 feet.
We can think of the slide as the hypotenuse of a right triangle, where:
- The height from the top of the slide to the ground is the opposite side.
- The angle at the top of the slide is 31.66°.
- The hypotenuse (the slide) is 46 feet.
We can use the sine function, which is defined as:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
So, we can express the height (opposite side) as:
\[ \text{height} = \text{hypotenuse} \cdot \sin(\theta) \]
Substituting in the known values:
\[ \text{height} = 46 \cdot \sin(31.66°) \]
Now, we will calculate the sine of 31.66° and then find the height:
\[ \sin(31.66°) \approx 0.5236 \quad (\text{using a calculator}) \]
Now, calculate the height:
\[ \text{height} = 46 \cdot 0.5236 \approx 24.0946 \text{ feet} \]
Rounding to the nearest foot, the height from the top of the slide to the ground is approximately:
\[ \text{height} \approx 24 \text{ feet} \]
Thus, the answer is 24 feet.