Question

a ladder leans against a house at a 60° angle to the ground.
if the ladder extends to a length of 166 inches, what is the height of the house, rounded to the nearest hundredth of an inch?
191.63
143.78
111.38
98.04

Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle where the ladder is the hypotenuse ($c = 166$ inches) and the height of the house ($h$) is the side opposite the $60^{\circ}$ angle. We use the sine function: $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.

Step2: Substitute the values into the sine formula

Given $\theta = 60^{\circ}$ and $c = 166$. Since $\sin60^{\circ}=\frac{\sqrt{3}}{2}$, and $\sin60^{\circ}=\frac{h}{166}$.
We can solve for $h$: $h = 166\times\sin60^{\circ}$.

Step3: Calculate the value

$h=166\times\frac{\sqrt{3}}{2}=83\sqrt{3}$. Using $\sqrt{3}\approx1.732$, then $h = 83\times1.732=143.756\approx143.76$.

Answer:

143.76