Question

a fire truck extends a 100 foot ladder toward a building at a 75° angle, to reach a window. how high is the window above the fire truck? round your answer to the nearest hundredth.

Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle where the ladder is the hypotenuse ($c = 100$ feet) and the height of the window above the fire - truck is the side opposite the given angle ($\theta=75^{\circ}$). We use the sine function: $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.

Step2: Solve for the opposite side

Let $h$ be the height of the window. Using the formula $\sin\theta=\frac{h}{c}$, we substitute $\theta = 75^{\circ}$ and $c = 100$. So $h = c\times\sin\theta$.
Since $\sin(75^{\circ})=\sin(45^{\circ}+30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659$.
Then $h = 100\times\sin(75^{\circ})$.
$h=100\times0.9659 = 96.59$ (rounded to the nearest hundredth).

Answer:

$96.59$