Question

To estimate the height of a mountain, two students find the angle of elevation from a point (at ground level) b = 800 meters from the base of the mountain to the top of the mountain is β = 59 ∘ . The students then walk a = 2550 meters straight back and measure the angle of elevation to now be α = 32 ∘ . If we assume that the ground is level, use this information to estimate the height of the mountain.

Answer 1

Answer:

1462 m

Step-by-step explanation:

Given:-

- The height of the mountain = h

- The distance from the base of mountain to first position, b = 800 m

- The angle of elevation of top of mountain from 1st position, β = 59°

- The amount of distance walked away from initial position, a = 2550 m

- The new angle of elevation of top of mountain, α = 32°

Find:-

If we assume that the ground is level, use this information to estimate the height of the mountain.

Solution:-

- We will sketch two triangles. First triangle would have a vertical height " h " that will denote the height of the mountain. Then a horizontal line " b " which is the initial position from the base of the mountain. The connect the ends of vertical and horizontal line by an hypotenuse. Forming and angle of elevation from first position to be β .

- Then we will use the trigonometric function of tangent to determine the height "h":

                             tan ( β ) = h1 / b

                             h1 = b*tan ( β )

                             h1 = 800*tan ( 59 )

                             h1 = 1331.42358 m

- Similarly, second triangle would have a vertical height " h " that will denote the height of the mountain. Then a horizontal line " a " which is the final position from the base of the mountain. The connect the ends of vertical and horizontal line by an hypotenuse. Forming and angle of elevation from final position to be α .

- Then we will use the trigonometric function of tangent to determine the height "h":

                             tan ( α ) = h2 / ( a )

                             h2 = (a)*tan ( β )

                             h2 = (2550)*tan ( 32 )

                             h2 = 1593.416 m

- To estimate the height of the mountain we will take an average of the two values obtained:

                            h_avg = ( h1 + h2 ) / 2

                                       = ( 1331.42358 + 1593.416 ) / 2

                                       = 1462.41979 m

Answer 2

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

Step-by-step 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

Therefore the height of the mountain is approximately 1,331.4 meters