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:
- \( a^2 + b^2 = c^2 \)
- \( b = 25 \) feet (hypotenuse)
- \( a = 8 \) 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°.