Question

Caleb wants to put up holiday lights on his house. He places the base of the 25-foot ladder at a 8-foot distance from the house. Using the inverse of sine, what is the approximate angle formed where the ladder rests on the house?(1 point)
Responses

40°
40°

19°
19°

11°
11°

30°

Answer 1

To find the angle formed where the ladder rests against the house, we can use the sine function. The situation can be visualized as a right triangle, where:

- The ladder represents the hypotenuse (\(h = 25\) ft).
- The horizontal distance from the base of the ladder to the house represents the adjacent side (\(a = 8\) ft).
- The angle we want to find (\(\theta\)) is between the ground and the ladder.

The sine function is defined as:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

In this case, we do not directly have the opposite side (the height at which the ladder rests on the house), but we can find the adjacent side since we need to find \(\theta\) using the cosine function instead:

\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{h} = \frac{8}{25}
\]

Now we can calculate \(\theta\) using the inverse cosine function:

\[
\theta = \cos^{-1}\left(\frac{8}{25}\right)
\]

Calculating \(\frac{8}{25}\):

\[
\frac{8}{25} = 0.32
\]

Now, calculate \(\theta\):

\[
\theta \approx \cos^{-1}(0.32)
\]

Using a calculator to find the inverse cosine:

\[
\theta \approx 72.34^\circ
\]

Since we are looking for the angle that the ladder makes with the ground, we can also find the complementary angle using:

\[
\text{angle with the house} = 90^\circ - \theta \approx 90^\circ - 72.34^\circ \approx 17.66^\circ
\]

Rounding this to the nearest response, the closest angle from the given choices is:

**19°**

Thus, the answer is:

**19°**