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 relationship

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

Step2: Set up the equation

Using \(\cos(58^\circ)=\frac{a}{75}\), we can solve for \(a\) by multiplying both sides by 75: \(a = 75\times\cos(58^\circ)\).

Step3: Calculate the value

First, find \(\cos(58^\circ)\) using a calculator. \(\cos(58^\circ)\approx0.5299\). Then multiply by 75: \(a = 75\times0.5299 = 39.7425\). Rounding to the nearest hundredth gives \(39.74\).

Answer:

39.74