Question

a fire truck extends a 90 foot ladder toward a building at a 61° angle, to reach a window. how far away is the fire truck from the building? ? feet round your answer to the nearest hundredth.

Explanation:

Step1: Identify right - triangle relationship

We have a right - triangle where the ladder is the hypotenuse ($c = 90$ feet) and the distance from the fire truck to the building is the adjacent side to the given angle $\theta=61^{\circ}$. We use the cosine function $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$.
Let the distance from the fire truck to the building be $x$. So, $\cos\theta=\frac{x}{c}$.

Step2: Solve for $x$

We know that $\theta = 61^{\circ}$ and $c = 90$. Substituting into the formula $\cos\theta=\frac{x}{c}$, we get $x = c\times\cos\theta$.
$x=90\times\cos(61^{\circ})$.
Since $\cos(61^{\circ})\approx0.4848$, then $x = 90\times0.4848 = 43.632$.

Step3: Round the answer

Rounding $43.632$ to the nearest hundredth, we get $43.63$.

Answer:

$43.63$