Question

a ladder leans against a house at a 60 - degree 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?

Explanation:

Step1: Identify the trigonometric relation

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

Step2: Substitute the values into the sine formula

$\sin(60^{\circ})=\frac{h}{166}$, where $h$ is the height of the house. Since $\sin(60^{\circ})=\frac{\sqrt{3}}{2}\approx0.866$.

Step3: Solve for $h$

$h = 166\times\sin(60^{\circ})=166\times0.866 = 143.756\approx143.76$ inches.

Answer:

143.76