Question

the ramp of a moving truck touches the ground 12 feet away from the end of the truck. if the ramp makes an angle of 19° relative to the ground, what is the length of the ramp? round your answer to two decimal places.

Explanation:

Step1: Identify the trigonometric relationship

We know the adjacent - side to the angle (\(x = 12\) feet) and we want to find the hypotenuse (\(r\)) of a right - triangle. The cosine function is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 19^{\circ}\) and the adjacent side to the angle \(\theta\) is the distance from the end of the truck to the point where the ramp touches the ground (\(a = 12\) feet), and the hypotenuse is the length of the ramp (\(l\)). So, \(\cos\theta=\frac{a}{l}\), and we can rewrite it as \(l=\frac{a}{\cos\theta}\).

Step2: Substitute the values

We know that \(a = 12\) feet and \(\theta=19^{\circ}\). First, find the value of \(\cos(19^{\circ})\). Using a calculator, \(\cos(19^{\circ})\approx0.9455\). Then, \(l=\frac{12}{\cos(19^{\circ})}\). Substituting the value of \(\cos(19^{\circ})\), we get \(l=\frac{12}{0.9455}\approx12.69\) feet.

Answer:

12.69 feet