Question

a fire truck extends a 75 foot ladder toward a building at a 58° angle to reach a window. how far away is the fire truck from the building? round your answer to the nearest hundredth.

Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle where the ladder is the hypotenuse ($c = 75$ feet) and the angle between the ladder and the ground is $\theta=58^{\circ}$, and we want to find the adjacent side $x$ (distance between the fire - truck and the building). We use the cosine function: $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$.
So, $\cos\theta=\frac{x}{c}$.

Step2: Substitute the values

Substitute $\theta = 58^{\circ}$ and $c = 75$ into the formula $\cos\theta=\frac{x}{c}$. We get $\cos(58^{\circ})=\frac{x}{75}$.

Step3: Solve for $x$

Multiply both sides of the equation by 75: $x = 75\times\cos(58^{\circ})$.
We know that $\cos(58^{\circ})\approx0.5299$, so $x = 75\times0.5299 = 39.7425$.

Step4: Round the answer

Rounding $39.7425$ to the nearest hundredth gives $x\approx39.74$.

Answer:

$39.74$