Question

you are building a ramp that must cover a horizontal distance of exactly 19 feet. the angle of the ramp from the ground is 13°. determine the length of the ramp, in feet. round to two decimal places as needed. show your work here

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the horizontal distance (adjacent side) is 19 feet, the angle with the ground is \(13^\circ\), and the length of the ramp is the hypotenuse (\(c\)). We use the cosine function, which is defined as \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). So, \(\cos(13^\circ)=\frac{19}{c}\).

Step2: Solve for the hypotenuse \(c\)

Rearrange the formula to solve for \(c\): \(c = \frac{19}{\cos(13^\circ)}\). First, calculate \(\cos(13^\circ)\). Using a calculator, \(\cos(13^\circ)\approx0.9743700648\). Then, \(c=\frac{19}{0.9743700648}\approx19.50\) (rounded to two decimal places).

Answer:

The length of the ramp is approximately \(\boxed{19.50}\) feet.