To find the angle between the ladder and the tree, we can use the sine function. The ladder forms a right triangle with the ground and the tree, where the ladder is the hypotenuse.
Let:
- \( a \) be the distance from the base of the ladder to the base of the tree, which is 8 feet.
- \( c \) be the length of the ladder, which is 15 feet.
- \( \theta \) be the angle between the ladder and the ground (or the tree).
The sine of the angle \( \theta \) is given by the ratio of the opposite side (the height of the ladder against the tree, which we can calculate using the Pythagorean theorem) to the hypotenuse:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{15} \]
To find \( h \) (the height), we use the Pythagorean theorem:
\[ h^2 + 8^2 = 15^2 \] \[ h^2 + 64 = 225 \] \[ h^2 = 225 - 64 \] \[ h^2 = 161 \] \[ h = \sqrt{161} \approx 12.6886 , \text{feet} \]
Now we can find \( \theta \) using the sine function:
\[ \sin(\theta) = \frac{h}{15} \] \[ \sin(\theta) = \frac{\sqrt{161}}{15} \]
Calculating this value:
\[ \sin(\theta) \approx \frac{12.6886}{15} \approx 0.8459 \]
Now, we can use the inverse sine function to find \( \theta \):
\[ \theta = \arcsin(0.8459) \]
Using a calculator to find \( \theta \):
\[ \theta \approx 58° \]
Thus, the approximate angle between the ladder and the tree is 58°.