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? feet round your answer to the nearest hundredth.

Explanation:

Step1: Identify the trigonometric ratio

We know the length of the hypotenuse (ladder length \(c = 75\) feet) and the angle \(\theta=58^{\circ}\). We need to find the adjacent side \(x\) (distance from fire - truck to building). Use the cosine ratio: \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\cos\theta=\frac{x}{c}\).

Step2: Solve for \(x\)

Substitute \(\theta = 58^{\circ}\) and \(c = 75\) into the formula \(x = c\times\cos\theta\). Then \(x=75\times\cos(58^{\circ})\).
We know that \(\cos(58^{\circ})\approx0.5299\).
So \(x = 75\times0.5299=39.7425\).

Answer:

\(39.74\)