Romeo owns a business that puts up and takes down holiday lights. He is working on a house and places the base of his 20-foot ladder at a 4-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)

Question

Responses

90°
90°

12°
12°

78°
78°

34°

Answer 1

To find the angle formed by the ladder resting against the house, we can use the sine function based on the right triangle formed. The ladder acts as the hypotenuse, the height of the ladder against the house is the opposite side, and the distance from the wall is the adjacent side.

Let’s denote:

• \( L \) = length of the ladder (hypotenuse) = 20 feet
• \( d \) = distance from the base of the ladder to the house (adjacent side) = 4 feet
• \( h \) = height the ladder reaches on the house (opposite side)

We can find the height \( h \) using the Pythagorean theorem:

\[ L^2 = h^2 + d^2 \] \[ 20^2 = h^2 + 4^2 \] \[ 400 = h^2 + 16 \] \[ h^2 = 400 - 16 = 384 \] \[ h = \sqrt{384} \approx 19.6 \text{ feet} \]

Now we can find the angle \( \theta \) formed by the ladder and the ground using the inverse sine function. The sine of the angle is given by the ratio of the opposite side to the hypotenuse:

\[ \sin(\theta) = \frac{h}{L} = \frac{\sqrt{384}}{20} \approx \frac{19.6}{20} \approx 0.98 \]

Now, we can find the angle \( \theta \):

\[ \theta = \sin^{-1}(0.98) \]

Calculating this gives approximately:

\[ \theta \approx 78.5° \]

Rounding this to the nearest whole number gives us approximately 78°.

So the answer is:

78°