Question
Deepa leans a 30-foot ladder against a wall. If the ladder reaches 29.5 feet up the wall, what angle does it form with the wall? Round your answer to the nearest tenth of a degree.
Answers
GPT-4o mini
To solve for the angle that the ladder forms with the wall, we can use the trigonometric function cosine. The relationship is given by:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
In this scenario, the adjacent side is the height reached by the ladder up the wall (29.5 feet) and the hypotenuse is the length of the ladder (30 feet). Thus, we can write:
\[
\cos(\theta) = \frac{29.5}{30}
\]
Calculating the value of \( \frac{29.5}{30} \):
\[
\cos(\theta) = \frac{29.5}{30} = 0.9833\overline{3}
\]
Now we will take the inverse cosine to find \( \theta \):
\[
\theta = \cos^{-1}(0.9833\overline{3})
\]
Using a calculator to find the angle:
\[
\theta \approx \cos^{-1}(0.9833\overline{3}) \approx 10.2^\circ
\]
Rounding to the nearest tenth of a degree, the angle that the ladder forms with the wall is:
\[
\boxed{10.2}
\] degrees.
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
In this scenario, the adjacent side is the height reached by the ladder up the wall (29.5 feet) and the hypotenuse is the length of the ladder (30 feet). Thus, we can write:
\[
\cos(\theta) = \frac{29.5}{30}
\]
Calculating the value of \( \frac{29.5}{30} \):
\[
\cos(\theta) = \frac{29.5}{30} = 0.9833\overline{3}
\]
Now we will take the inverse cosine to find \( \theta \):
\[
\theta = \cos^{-1}(0.9833\overline{3})
\]
Using a calculator to find the angle:
\[
\theta \approx \cos^{-1}(0.9833\overline{3}) \approx 10.2^\circ
\]
Rounding to the nearest tenth of a degree, the angle that the ladder forms with the wall is:
\[
\boxed{10.2}
\] degrees.