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 was 25 degrees. find the distance from the student to the point on level ground directly beneath the peak. (round your answer to one decimal place.)

Explanation:

Step1: Recall the tangent function

The tangent of an angle in a right - triangle is given by \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Let the height of the mountain (opposite side) be \(h = 12400\) feet and the angle of elevation \(\theta=20^{\circ}\), and the distance from the student to the point beneath the peak (adjacent side) be \(x\). So, \(\tan\theta=\frac{h}{x}\).

Step2: Solve for \(x\)

We can rewrite the formula as \(x=\frac{h}{\tan\theta}\). Substitute \(h = 12400\) and \(\theta = 20^{\circ}\). Since \(\tan(20^{\circ})\approx0.3640\), then \(x=\frac{12400}{0.3640}\).

Step3: Calculate the value of \(x\)

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

Answer:

\(34065.9\)