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°

11°
11°

30°
30°

19°

Answer 1

To find the angle formed where the ladder rests against the house, we can use the inverse sine function. We can visualize the situation as a right triangle where:

- The length of the ladder (hypotenuse) = 25 feet
- The distance from the base of the ladder to the house (adjacent side) = 8 feet
- The height at which the ladder reaches the house (opposite side) can be found using the Pythagorean theorem.

First, we calculate the height using the Pythagorean theorem:

1. \( a^2 + b^2 = c^2 \)
2. \( b = 25 \) feet (hypotenuse)
3. \( a = 8 \) feet (the distance from the wall)

Let \( h \) be the height the ladder reaches on the house (opposite side).

Using \( 8^2 + h^2 = 25^2 \):

\[
8^2 + h^2 = 25^2
\]
\[
64 + h^2 = 625
\]
\[
h^2 = 625 - 64
\]
\[
h^2 = 561
\]
\[
h = \sqrt{561} \approx 23.7 \text{ feet}
\]

Now, we can find the angle \( \theta \) at which the ladder rests using the cosine function:

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

To find \( \theta \):

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

Calculating \( \cos^{-1} \) using a calculator:

\[
\theta \approx 69.7^\circ
\]

Now, using the sine function:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{25}
\]

To find the angle \( \phi \):

\[
\phi = \sin^{-1}\left(\frac{h}{25}\right) = \sin^{-1}\left(\frac{\sqrt{561}}{25}\right)
\]

Calculating this gives us approximately:

\[
\phi \approx 48.2^\circ
\]

Given that we are asked for possible options which likely use either sine or cosine.

For options given:
- 40°
- 11°
- 30°
- 19°

If we want to use sine as calculated for the setup for \( \tan(\theta) = \frac{h}{8} \)

Given the inputs, we are looking for a rough estimate; \( 30° \) seems the closest and more reasonable than other options.

**Final selected answer: 30°**.