Question

the peak of mt. fuji in japan is approximately 12,400 feet high. a trigonometry student, several miles away, notes that the angle between level ground and the peak is 20°. estimate the distance from the student to the point on level ground directly beneath the peak. (round your answer to one decimal place.)

Explanation:

Step1: Use the tangent function

We know that in a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 20^{\circ}\), the opposite side (height of the peak) \(h = 12400\) feet, and the adjacent side \(x\) is the distance from the student to the point on level ground directly beneath the peak. So, \(\tan(20^{\circ})=\frac{12400}{x}\).

Step2: Solve for \(x\)

Rearrange the formula to get \(x=\frac{12400}{\tan(20^{\circ})}\). We know that \(\tan(20^{\circ})\approx0.3640\). Then \(x=\frac{12400}{0.3640}\).

Step3: Calculate the value of \(x\)

\(x=\frac{12400}{0.3640}\approx34065.9\) feet.

Answer:

\(34065.9\) feet