a ramp leading into a building makes a 15° angle with the ground. the end of the ramp is 10 feet from the…

a ramp leading into a building makes a 15° angle with the ground. the end of the ramp is 10 feet from the base of the building (along the ground). approximately how long is the ramp? round to the nearest tenth. 10.3 feet 10.4 feet 13.2 feet 38.6 feet

a ramp leading into a building makes a 15° angle with the ground. the end of the ramp is 10 feet from the base of the building (along the ground). approximately how long is the ramp? round to the nearest tenth. 10.3 feet 10.4 feet 13.2 feet 38.6 feet

Answer

Explanation:

Step1: Identify the trig - ratio

We know the adjacent side to the angle and want to find the hypotenuse. We use the cosine function. $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 15^{\circ}$ and the adjacent side $x = 10$ feet, and the hypotenuse is the length of the ramp $l$. So, $\cos15^{\circ}=\frac{10}{l}$.

Step2: Solve for $l$

We can re - arrange the formula to get $l=\frac{10}{\cos15^{\circ}}$. We know that $\cos15^{\circ}=\cos(45^{\circ}- 30^{\circ})=\cos45^{\circ}\cos30^{\circ}+\sin45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659$. Then $l=\frac{10}{0.9659}\approx10.4$ feet.

Answer:

10.4 feet