Question

Please help!!! GEOMETRY 20 POINTS

Answer 1

Answer: The height of the mountain is 1,331.4 meters (approximately)

Explanation:

From the information given, the students were standing at point b which is 800 meters from the base of the mountain and the angle of elevation from that point is 59°. Assuming that the ground is level, we can derive a right angled triangle from this set of details and hence we have triangle ABC, where angle β is the reference angle, (59 degrees), BC is the distance from the students to the base of the mountain (800 meters) and the line AC is the height of the mountain.

The line AC is the opposite, since angle B is the reference angle, therefore we shall use the trigonometric ratio as follows;

Tan β = opposite/adjacent

Tan 59 = AC/800

Tan 59 x 800 = AC

1.6643 x 800 = AC

1331.44 = AC

AC ≈ 1331.4

Answer 2

h = 1886.8 m

Explanation:

Step 1: Get one more angle in the left triangle.

Notice that there is an angle suplementary to angle β.

angle = 180° - β = 180°- 49° = 131°

Step 2: Get hypotenuse.

You can either calculate hypotenuse of the big right triangle or small right triangle.

We know two angles in the left triangle - angle α and the angle calculated in previous step. We also know that distance a is 1500 m. We can use sine law to get the hypotenuse. I chose to calculate the hypotenuse of the big right triangle (big right triangle is the one that includes both left and right little triangle).

We also have to find the angle that is opposite of side a.

opp_a_angle = 180° - α - angle = 180° - 31°- 131° = 18°

`(a)/(sin((opp\_a\_angle))) = (hypotenuse)/(sin((angle)))`

`(1500)/(sin((18^\circ))) = (hypotenuse)/(sin((131^\circ)))`

`hypotenuse = (1500 \cdot sin((131^\circ)))/(sin((18^\circ)))`

`hypotenuse \approx 3663.437257 \text{ m}`

Step 3: Calculate height

To calculate height we can use trigonometric functions. In this case it's going to be sine.

Formula:

`sin(\theta) = (opposite)/(hypotenuse)`

`sin(\alpha) = (h)/(hypotenuse)`

`sin((31^\circ)) = (h)/(3663.437257)`

`1886.8096 \approx h`

`h = 1886.8 \text{ m}`