A helicopter is flying at a height of 1000 feet above a small mountain peak as show in the figure below. The smaller mountain peak has an altitude of 6200 feet. A second, taller peak is viewed from both the smaller mountaintop and the helicopter. From the helicopter, the angle of depression is 43°, and from the smaller mountaintop, the angle of depression is 18°. Approximate the altitude of the taller peak and round your answer to the nearest foot.
check your wording:
... from the smaller mountaintop, the angle of depression is 18°
If you are looking from the smaller to the larger, it would be an angle of elevation, not depression.
I will assume that case.
Draw a diagram, draw a horizontal from the taller mountain
cutting the 1000 ft into x and 1000-x, the x close to the small mountain top
Let that horizontal distance be h
You now have two right-angled triangles so that
h/x = cot43° and h/(1000-x) = cot 18°
h = xcot43 and h = (1000-x)cot18
xcot43 = (1000-x)cot18
xcot43 + xcot18 = 1000
x = 1000/(cot43 + cot18) = ..... use your calculator, remember to get cotx, do 1/tanx
Look at your diagram, what does the x represent?
To solve this problem, we can use trigonometry and the concept of angle of depression.
Let's start by drawing a diagram to better understand the situation.
In the diagram, we have a smaller mountain peak, the helicopter, and a taller peak. The helicopter is flying at a height of 1000 feet above the smaller mountain peak. The smaller mountain peak has an altitude of 6200 feet.
Now, let's label the unknown altitude of the taller peak as "x" feet.
From the helicopter's perspective, we have an angle of depression of 43°. This means that if we look straight down from the helicopter, we will see a line forming a 43° angle with a horizontal line.
Similarly, from the smaller mountain peak's perspective, we have an angle of depression of 18°.
Based on this information, we can set up two separate right triangles to represent the perspectives of the helicopter and the smaller mountain peak.
First, let's look at the triangle formed from the helicopter's perspective.
We have a right triangle with the altitude of the taller peak (x feet) as the opposite side, the distance between the helicopter and the smaller mountain peak (1000 feet) as the adjacent side, and the hypotenuse as the line of sight from the helicopter to the taller peak.
Using trigonometry, we can apply the tangent function to this triangle:
tan(43°) = x / 1000
Next, let's look at the triangle formed from the smaller mountain peak's perspective.
We have a right triangle with the altitude of the taller peak (x feet + 6200 feet) as the opposite side, the distance between the smaller mountain peak and the taller peak (unknown) as the adjacent side, and the hypotenuse as the line of sight from the smaller mountain peak to the taller peak.
Applying the tangent function to this triangle, we have:
tan(18°) = (x + 6200) / unknown
Now, we can solve these two equations simultaneously to find the value of x, which represents the altitude of the taller peak.
By isolating x in the first equation, we get:
x = tan(43°) * 1000
Plugging in the value of tan(43°) into a calculator, we have:
x ≈ 1354.232 feet
Now, let's plug the value of x into the second equation:
tan(18°) = (1354.232 + 6200) / unknown
Rearranging the equation to find the value of unknown, we have:
unknown = (1354.232 + 6200) / tan(18°)
Plugging in the values into a calculator, we have:
unknown ≈ 19195.668 feet
However, this value represents the total distance from the smaller mountain peak to the taller peak. We want to find the altitude of the taller peak. Therefore, we subtract the altitude of the smaller mountain peak (6200 feet) from this value:
altitude of the taller peak = 19195.668 - 6200 ≈ 12995.668 feet
Rounding this value to the nearest foot as requested, the approximate altitude of the taller peak is 12996 feet.