Question

A ladder leans against a house at a 60° angle to the ground.

House with a ladder measuring 132 inches forming a right triangle. The base angle formed by the ladder is 60 degrees.

If the ladder extends to a length of 132 inches, what is the height of the house, rounded to the nearest hundredth of an inch?

76.21
93.34
114.32
186.68

Answer 1

To find the height of the house using the ladder's length and the angle it makes with the ground, we can use the sine function from trigonometry.

In this scenario, the ladder forms a right triangle where:
- The hypotenuse is the length of the ladder (132 inches).
- The angle between the ladder and the ground is 60°.
- The height of the house, which is the vertical side of the triangle, is what we're trying to find.

The sine function relates the angle to the opposite side (height of the house) and hypotenuse (length of the ladder) as follows:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

Rearranging this equation to solve for the opposite side (height):
\[
\text{height} = \text{hypotenuse} \times \sin(\theta)
\]
Substituting in the values we have:
\[
\text{height} = 132 \times \sin(60^\circ)
\]

From trigonometry, we know that:
\[
\sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866025
\]

Now we can compute the height:
\[
\text{height} = 132 \times 0.866025 \approx 114.32
\]

Therefore, the height of the house, rounded to the nearest hundredth of an inch, is:
\[
\boxed{114.32} \text{ inches}
\]