To find the angle formed where the ladder rests against the house, we can use the sine function, which relates the opposite side of a right triangle to the hypotenuse. In this case, the height that the ladder reaches against the house is the opposite side, and the length of the ladder is the hypotenuse.
- The ladder length (hypotenuse) = 20 feet
- The distance from the base of the ladder to the house (adjacent side) = 4 feet
We can use the following relationship:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Where the opposite side can be found using the Pythagorean theorem:
\[ \text{opposite} = \sqrt{\text{hypotenuse}^2 - \text{adjacent}^2} \] \[ \text{opposite} = \sqrt{20^2 - 4^2} \] \[ = \sqrt{400 - 16} = \sqrt{384} \approx 19.6 \text{ feet} \]
Now we can calculate the angle using:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{19.6}{20} \] \[ \theta = \arcsin\left(\frac{19.6}{20}\right) \]
Calculating this gives:
\[ \theta \approx \arcsin(0.98) \approx 78.5° \]
Rounding to the nearest degree gives us about 78°.
So the answer is:
78°