To find the angle between the ladder and the tree, we can use the sine function. The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
In this scenario, we have:
- The length of the ladder (the hypotenuse) is 15 feet.
- The distance from the base of the ladder to the tree (the base of the triangle) is 8 feet.
- We need to determine the height of the point where the ladder touches the tree (the opposite side).
First, we can find the height using the Pythagorean theorem:
\[ h = \sqrt{15^2 - 8^2} = \sqrt{225 - 64} = \sqrt{161} \approx 12.6886 \text{ feet} \]
Now, we can find the angle \( \theta \) between the ladder (the hypotenuse) and the tree (the opposite side) using the sine function:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{15} \] \[ \sin(\theta) = \frac{\sqrt{161}}{15} \]
Now, calculate \( \theta \) using the inverse sine (arcsin):
\[ \theta \approx \arcsin\left(\frac{\sqrt{161}}{15}\right) \]
Using a calculator to find the value:
\[ \theta \approx \arcsin\left(\frac{12.6886}{15}\right) \approx \arcsin(0.8459) \approx 57.59^\circ \]
Thus, the approximate angle between the ladder and the tree is:
\[ \theta \approx 57.59^\circ \]
Therefore, the response is approximately 57.6 degrees.
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