Question

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

Explanation:

Step1: Identify the trig - ratio

We know the adjacent side to the angle ($x = 10$ feet) and we want to find the hypotenuse ($h$) of the right - triangle formed by the ramp, the ground, and the building. We use the cosine function since $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 15^{\circ}$ and the adjacent side to the $15^{\circ}$ angle is 10 feet. So, $\cos(15^{\circ})=\frac{10}{h}$.

Step2: Solve for $h$

We can re - arrange the formula $\cos(15^{\circ})=\frac{10}{h}$ to get $h=\frac{10}{\cos(15^{\circ})}$. We know that $\cos(15^{\circ})=\cos(45^{\circ}- 30^{\circ})=\cos45^{\circ}\cos30^{\circ}+\sin45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659$. Then $h=\frac{10}{0.9659}\approx10.4$.

Answer:

10.4 feet