Question

Hello there.
And I have a little problem about sine law problems.
And I seem I can't get the right answer.
So here's the question:
As you walk on a straight level path toward a mountain, the measure of the angle of elevation to the peak of the mountain from one point is 33 degrees. From a point 1000 ft closer, the angle of elevation is 35 degrees. How high is the mountain?

Thanks in advance.

Answer 1

Answer on Question #49749 - Math - Trigonometry

As you walk on a straight level path toward a mountain, the measure of the angle of elevation to the peak of the mountain from one point is 33 degrees. From a point 1000 ft closer, the angle of elevation is 35 degrees. How high is the mountain?

Solution:

$$\angle B A D = 3 5 {}^ {\circ}, \angle B C A = 3 3 {}^ {\circ}, A C = 1 0 0 0$$

Let $h$ be the height of the mountain. Let's consider the triangle $ABC$ . The sum of angles $\angle ABC = \hat{B}$ and $\angle ACB = \hat{C}$ equals the value of angle $\angle BAD$ , because $\hat{A} + \hat{B} + \hat{C} = 180{}^\circ$ , where $\hat{A} = \angle BAC$ , $\hat{A} + \angle BAD = 180{}^\circ$ , hence $\angle BAD = 180{}^\circ - \hat{A} = \hat{B} + \hat{C}$ .

From $\angle BAD = \hat{B} +\hat{C}$ we easily obtain that $\hat{B} = \angle BAD - \hat{C} = 35{}^{\circ} - 33{}^{\circ} = 2{}^{\circ}$ . Using the law of sines for triangle $ABC$ we obtain

$$\frac {A C}{\sin \hat {B}} = \frac {B A}{\sin \hat {C}}$$

Solving this equation for $BA$ we obtain

$$B A = A C \frac {\sin \hat {C}}{\sin \hat {B}}$$

Now consider the triangle $ABD$ . Using the law of sines for this triangle we obtain

$$\frac {B D}{\sin \angle B A D} = \frac {B A}{\sin \widehat {D}}$$

Solving this equation for $BD$ and considering the fact that $\sin \widehat{D} = 1$ ( $\widehat{D} = 90{}^\circ$ ) we obtain

$$B D = B A \sin \angle B A D$$

Substitution of $BA$ from the equation (1) gives us the following

$$h = BD = BA \sin \angle BAD = AC \frac{\sin \hat{C}}{\sin \hat{B}} \sin \angle BAD = 1000 \text{ ft} \cdot \frac{\sin 33{}^\circ}{\sin 2{}^\circ} \sin 35{}^\circ = 8951 \text{ ft}$$

Answer: 8951 ft.

www.AssignmentExpert.com