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 trigonometric relationship

We know the adjacent side (\(x = 10\) feet) to the angle (\(\theta=15^{\circ}\)) and we need to find the hypotenuse (\(r\)) (length of the ramp). Use the cosine formula \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\cos(15^{\circ})=\frac{10}{r}\).

Step2: Solve for \(r\)

Rearrange the formula to \(r = \frac{10}{\cos(15^{\circ})}\).
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 \(r=\frac{10}{0.9659}\approx10.4\).

Answer:

10.4 feet