Question

a survey team is trying to estimate the height of a mountain above a level plain. From one point on the plain they observed at the angle of elevation to the top of the mountain is 26 degrees. From a point 1500 feet closer to the mountain along the plain, they find that the angle of elevation is 30 degrees. How high in feet is the mountain?

Answer 1

4713.23 ft

Explanation:

A sketch of the problem is given below:

• The survey team was initially at point A above, the angle of elevation is 26 degrees.

,

• They moved 1500 feet closer to point B and the angle of elevation is 30 degrees.

We want to find the height, h of the mountain.

Recall from trigonometry:

`\tan \theta=\frac{\text{Opposite}}{\text{Adjacent}}`

In right triangle EFB:

`\begin{gathered} \tan B=(EF)/(FB) \\ \tan 30\degree=(h)/(x) \\ \implies h=x\tan 30\degree \end{gathered}`

Likewise, in right triangle EFA:

`\begin{gathered} \tan A=(EF)/(FA) \\ \tan 26\degree=(h)/(x+1500) \\ \implies h=(x+1500)\tan 26\degree \end{gathered}`

Equate the heights from the two obtained above:

`x\tan 30\degree=(x+1500)\tan 26\degree`

Then, solve the equation for x.

Finally, substitute x=8163.56 ft to solve for h, the height.

`\begin{gathered} h=x\tan 30\degree \\ =8163.5552*\tan 30\degree \\ =4713.23ft \end{gathered}`

The height of the mountain is 4713.23 ft correct to 2 decimal places.