To find the angle formed by the ladder resting against the house, we can use the sine function in a right triangle. In this case, we have:
- The length of the ladder (the hypotenuse) = 20 feet
- The distance from the base of the ladder to the house (one leg of the right triangle) = 4 feet
To find the angle \( \theta \) where the ladder rests against the house, we can use the following relationship in a right triangle:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Here, the "opposite" side is the height the ladder reaches on the house, which we need to calculate.
First, we will find the height of the ladder using the Pythagorean theorem:
\[ \text{height}^2 + \text{base}^2 = \text{hypotenuse}^2 \]
Let the height be \( h \):
\[ h^2 + 4^2 = 20^2 \] \[ h^2 + 16 = 400 \] \[ h^2 = 400 - 16 \] \[ h^2 = 384 \] \[ h = \sqrt{384} \approx 19.6 \text{ feet} \]
Next, we can now find the angle using the sine function. The value of \( \sin(\theta) \) will be:
\[ \sin(\theta) = \frac{h}{20} = \frac{19.6}{20} \]
Now using the inverse sine (arcsin) function to find \( \theta \):
\[ \theta = \arcsin\left(\frac{19.6}{20}\right) \approx \arcsin(0.98) \]
Calculating the angle using a calculator:
\[ \theta \approx 78° \]
Thus, the approximate angle formed where the ladder rests on the house is:
78°.