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 have a right triangle where the adjacent side to the \(15^\circ\) angle is 10 feet (distance from ramp end to building base), and the ramp is the hypotenuse (\(c\)) we need to find. We use the cosine function, which is \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). So, \(\cos(15^\circ)=\frac{10}{c}\).

Step2: Solve for the hypotenuse \(c\)

Rearrange the formula to solve for \(c\): \(c = \frac{10}{\cos(15^\circ)}\). Calculate \(\cos(15^\circ)\approx0.9659\). Then \(c=\frac{10}{0.9659}\approx10.35\). Rounding to the nearest tenth gives \(10.4\) (wait, no, wait: wait, \(10\div\cos(15^\circ)\): let's recalculate. \(\cos(15^\circ)\) is approximately \(0.9659258263\). So \(10\div0.9659258263\approx10.35\), which rounds to \(10.4\)? Wait, no, wait, maybe I made a mistake. Wait, no, let's check again. Wait, the adjacent side is 10, angle with ground is \(15^\circ\), so cosine of angle is adjacent over hypotenuse. So hypotenuse is adjacent over cos(angle). So \(10 / \cos(15^\circ)\). Let's compute \(\cos(15^\circ)\): \(\cos(15^\circ)=\cos(45^\circ - 30^\circ)=\cos45\cos30+\sin45\sin30=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx\frac{2.449 + 1.414}{4}\approx\frac{3.863}{4}\approx0.96575\). So \(10\div0.96575\approx10.35\), which is approximately \(10.4\) when rounded to the nearest tenth? Wait, but let's check with calculator: \(10\div\cos(15^\circ)\). Let's use calculator: \(\cos(15^\circ)\approx0.9659\), so \(10\div0.9659\approx10.35\), which is \(10.4\) when rounded to the nearest tenth? Wait, but the options have 10.3, 10.4, 13.2, 38.6. Wait, maybe I mixed up adjacent and hypotenuse? Wait, no, the ramp is the hypotenuse, the ground distance is adjacent. Wait, maybe it's cosine, but let's check again. Wait, angle with ground is \(15^\circ\), so the triangle: angle at ground is \(15^\circ\), adjacent side (ground) is 10, hypotenuse is ramp. So \(\cos(15^\circ)=\frac{10}{ramp}\), so ramp \(=\frac{10}{\cos(15^\circ)}\approx10.35\), which is \(10.4\) when rounded to nearest tenth. Wait, but let's check with sine: if we use sine, opposite side is height, but we don't know height. So cosine is correct. So the calculation gives approximately 10.4? Wait, but let's check with calculator: 10 divided by cos(15 degrees). Let's compute cos(15°):

Using calculator, cos(15°) ≈ 0.9659258263

10 / 0.9659258263 ≈ 10.3527618

Rounded to the nearest tenth is 10.4. So the answer is 10.4 feet.

Answer:

10.4 feet (corresponding to the option: 10.4 feet)