To find the angle between the ladder and the tree using the inverse sine function (also known as arcsine), we can use the relationship of a right triangle.
In this scenario:
- The ladder acts as the hypotenuse (c), which is 33 feet.
- The distance from the base of the ladder to the tree (the adjacent side) is 9 feet.
We can use the sine function, which is defined as:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
However, in this case, we know the adjacent side, and we need to find the angle using the hypotenuse and the opposite side. First, we find the height (the opposite side) using the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
Where:
- \(c\) is the hypotenuse (33 feet)
- \(a\) is the opposite side (height against the tree)
- \(b\) is the base (9 feet)
Rearranging, we can solve for the opposite side:
\[ 33^2 = a^2 + 9^2 \] \[ 1089 = a^2 + 81 \] \[ a^2 = 1089 - 81 \] \[ a^2 = 1008 \] \[ a \approx \sqrt{1008} \approx 31.7 \text{ feet} \]
Next, to find the angle θ, we can use the cosine function instead:
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{9}{33} \]
Now applying the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{9}{33}\right) \] \[ \theta = \cos^{-1}(0.2727) \approx 74.67° \]
To find the angle between the ladder and the tree, which is the supplementary angle:
\[ \text{Angle} = 90° - \theta \approx 90° - 74.67° \approx 15.33° \]
Now rounding to the nearest option, the closest angle is 16°.
Thus, the answer is:
16°.
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