Question

A roofer props a ladder against a wall so that the base of the ladder is 4 feet away from the building. If the angle of elevation from the bottom of the ladder to the roof is 63 degree, how long is the ladder?

a. 12.4 feet
b. 10 feet
c. 9.22 feet
d. 8.81 feet

Answer 1

To find the length of the ladder, we can use trigonometry, specifically the sine function, since we are given the angle of elevation and the distance from the wall to the base of the ladder.

We have the following information:
- The angle of elevation \( \theta = 63^\circ \)
- The distance from the wall (base of the ladder) \( b = 4 \) feet.

We want to find the length of the ladder \( L \).

Using the tangent function, which relates the angle of elevation to the opposite side (height of the roof) and the adjacent side (distance from the wall):
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Which implies:
\[
\text{opposite} = \tan(63^\circ) \cdot b
\]
However, since we need the length of the ladder, we can also use the cosine function:
\[
\cos(\theta) = \frac{\text{adjacent}}{L}
\]
or rearranging gives:
\[
L = \frac{\text{adjacent}}{\cos(\theta)} = \frac{b}{\cos(63^\circ)}
\]

Now substituting the known values:
\[
L = \frac{4}{\cos(63^\circ)}
\]

Using \(\cos(63^\circ) \approx 0.454\):
\[
L = \frac{4}{0.454} \approx 8.81 \text{ feet}
\]

Thus, the length of the ladder is approximately \( 8.81 \) feet.

The correct answer is **d. 8.81 feet**.