To find the angle between the ladder and the tree, we can use the sine function. In this scenario, we can visualize the situation as a right triangle where:
- The ladder (15 feet) is the hypotenuse.
- The distance from the base of the ladder to the base of the tree (8 feet) is the adjacent side.
- The height at which the ladder touches the tree is the opposite side.
First, we can find the height (opposite side) of the triangle using the Pythagorean theorem:
\[ h = \sqrt{(15^2) - (8^2)} = \sqrt{225 - 64} = \sqrt{161} \approx 12.688 \]
Now we can find the angle (let's call it θ) between the ladder and the ground using the sine function:
\[ \sin(θ) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{h}{15} \]
But since we want the angle between the ladder and the tree, we can instead use cosine, as the adjacent side is the distance from the bottom of the ladder to the tree:
\[ \cos(θ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{8}{15} \]
Now we can calculate the angle:
\[ θ = \cos^{-1}\left(\frac{8}{15}\right) \]
Calculating \( \frac{8}{15} \):
\[ \frac{8}{15} \approx 0.5333 \]
Now, using a calculator to find \( θ \):
\[ θ \approx \cos^{-1}(0.5333) \approx 58° \]
Thus, the approximate angle between the ladder and the tree is 58°.
So the correct response is 58°.