Question

question 19 of 26
a portion of a hiking trail slopes downward at about an 84° angle. to the nearest tenth of a foot, what is the value of x, the distance the hiker traveled along the path, if he has traveled a horizontal distance of 120 feet?
image of right triangle with right angle at a, angle at b is 84°, ac = 120 ft, bc = x

a. 130.2 ft
b. 143.3 ft
c. 120.7 ft
d. 12.5 ft

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle \( \triangle ABC \) with \( \angle A = 90^\circ \), \( \angle B = 84^\circ \), the adjacent side to \( \angle B \) is \( AB \) (vertical side), the opposite side is \( AC = 120 \) ft (horizontal side), and the hypotenuse is \( BC = x \) (the path traveled). We use the cosine function because \( \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \), but wait, actually, \( \cos(\angle C)=\frac{AC}{BC} \), but first, find \( \angle C \). Since the sum of angles in a triangle is \( 180^\circ \), \( \angle C=180^\circ - 90^\circ - 84^\circ = 6^\circ \). Wait, no, better to use \( \cos(84^\circ)=\frac{AB}{x} \) and \( \sin(84^\circ)=\frac{AC}{x} \). Wait, \( AC \) is the horizontal distance, which is the adjacent side to \( \angle B \)? Wait, no, \( \angle B = 84^\circ \), so the angle at \( B \), so the sides: \( AC \) is opposite to \( \angle B \), \( AB \) is adjacent to \( \angle B \), and \( BC = x \) is the hypotenuse. So \( \sin(\angle B)=\frac{AC}{BC} \), so \( \sin(84^\circ)=\frac{120}{x} \).

Step2: Solve for \( x \)

From \( \sin(84^\circ)=\frac{120}{x} \), we can rearrange to \( x = \frac{120}{\sin(84^\circ)} \). Calculate \( \sin(84^\circ) \approx 0.9945 \). Then \( x=\frac{120}{0.9945}\approx 120.7 \) ft? Wait, no, wait, maybe I mixed up the angle. Wait, the angle at \( B \) is \( 84^\circ \), so the angle at \( C \) is \( 6^\circ \). Let's check with cosine of angle \( C \). \( \cos(6^\circ)=\frac{AC}{BC} \), so \( BC = \frac{AC}{\cos(6^\circ)} \). \( \cos(6^\circ)\approx 0.9945 \), so \( x=\frac{120}{0.9945}\approx 120.7 \) ft. Wait, but let's recalculate. Wait, \( \sin(84^\circ)=\sin(90^\circ - 6^\circ)=\cos(6^\circ)\approx 0.9945 \). So \( x = \frac{120}{\sin(84^\circ)} \approx \frac{120}{0.9945219} \approx 120.7 \) ft. Wait, but let's check the options. Option C is 120.7 ft. Wait, but maybe I made a mistake. Wait, the angle is at \( B \), 84 degrees, so the horizontal distance is \( AC = 120 \) ft, which is adjacent to angle \( C \), and \( x \) is the hypotenuse. So angle at \( C \) is \( 6^\circ \), so \( \cos(6^\circ)=\frac{AC}{x} \), so \( x = \frac{AC}{\cos(6^\circ)} \). \( \cos(6^\circ) \approx 0.9945 \), so \( x \approx 120 / 0.9945 \approx 120.7 \). So the answer is C.

Wait, but let's check with calculator. Let's compute \( \sin(84^\circ) \): \( \sin(84) \approx 0.9945219 \). Then \( 120 / 0.9945219 \approx 120.66 \), which rounds to 120.7. So yes, that's correct.

Answer:

C. 120.7 ft