approximate the distance from peak to peak to the nearest foot.

Question

a helicopter hovers at an altitude that is 1142 feet above a mountain peak of altitude of 5820. A second, taller peak is viewed form both the mountaintop and helicopter. From the helicopter the angle of depression is 43 degrees, and from the mountaintop, the angle of elevation is 18 degrees.

Answer 1

Hello Morton,

 

The first challenge of this problem is to draw a representation of the description. As I am not able to show the drawing, please use your imagination and try to visualize the following: 

 

l>l

Draw horizontal lines across the top and bottom so that it looks like a rectangle and have all the segments touching one another, forming three triangles inside the rectangle. It's a stretch, but the best I can describe. 

 

The height of the vertical line on the left is 1142. The helicopter is hovering there at the top. The diagonal increasing from left to right is the distance between the peaks that we are trying to find, connecting the peak of the first mountain on the left to the peak of the taller, second mountain on the right. Please label this as x.

 

Let's list how things are labelled:

 

The vertical line on the left = 1142

The diagonal increasing from left to right = x

The width of the rectangle, the horizontal lines, both the top and bottom = w

 

Now, let's think about the vertical line on the right. Because this is a rectangle, this distance is also 1142, but it is made up of two smaller segments. 

 

The height of the top portion of the vertical line on the right = h

The height of the bottom portion of the vertical line on the right = 1142 - h

 

Still with me? 

 

Both the upper right hand corner of the rectangle and the lower right hand corner of the rectangle are right angles. Could you please indicate this in your figure? 

 

Let's get that 18 degree angle of elevation in there. Go to the bottom left hand side of the rectangle and label the angle from the "ground up", formed by the bottom of the rectangle (w) and the diagonal increasing from left to right (x).

 

Let's get that 43 degree angle of depression in there. Go to the upper left hand side of the rectangle and label the angle from the "sky down", formed by the top of the rectangle (also w) and the diagonal that is decreasing from left to right that we did not label.

 

Using the right triangle on the bottom, and given that tangent is defined as the ratio of the opposite side to the adjacent side, write the following equation:

 

tan 18 = (1142 - h) / w. Rewrite this, solving for w, as: 

 

w = (1142 -h) / tan 18.

 

Using the right triangle at the top, write the following equation:

 

tan 43 = h / w. Rewrite this, solving for w, as:

 

w = h / tan 43.

 

Set the two equations (that equal w) equal to each other:

 

(1142 - h) / tan 18 = h / tan 43

 

Cross multiply and set the products equal to each other. Be sure that your calculator mode is in degrees.

 

h tan 18 = tan 43 (1142 - h)

h tan 18 = 1142 tan 43 - h tan 43

h tan 18 + h tan 43 = 1142 tan 43

h (tan 18 + tan 43) = 1142 tan 43

h = (1142 tan 43) / (tan 18 + tan 43)

h ≈ 847 feet

 

Go back to your rectangle. On the right hand side, the top portion of the vertical line that we called h, now is determined to be 847 and the bottom portion of the vertical line that we called 1142 - h is now 295.

 

We're on the home stretch!

 

Using the bottom right triangle, and the fact that sin is defined as the ratio of the opposite side to the hypotenuse, write the following equation:

 

sin 18 ≈ 295 / x

 

Rewrite, solving for x, as:

 

x ≈ 295 / sin 18

 

x ≈ 955

 

The approximate distance between the two peaks is 955 feet. 

 

I hope this helps!