Angle of elevation- Math Central

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Question from Rita, a student:

Uluru or Ayers Rock is a sacred place for Aborigines of the western desert of Australia.
Chun-Wei uses a surveying device to measure the angle of elevation to the top of the rock to be 11.5 degrees.
He walks half a mile closer and measures the angle of elevation to be 23.9 degrees.
How high is Ayers Rock in feet?

This is my set:

y = height of Ayers Rock (in feet)

x = the total distance in feet from the point of the first measurement to the point directly under the highest point of Uluru.

tan11.5 degrees = y/x

tan23.9 degrees = y/x - 2640 feet

This is where I get stuck. Can someone perform the algebra part and find the answer?
Also, is there an easier method for answering similar questions?

Hi Rita. You are setting up the solution properly. Since it is y you need to find, you need to eliminate x from the two equations. When you solve each equation for x, you get

x = y/tan11.5
x = 2640 + y/tan23.9

Two things that both equal the same third thing equal each other, so this means

y/tan11.5 = 2640 + y/tan23.9

Solving this for y gives:

y = 2640 / ( cot 11.5 - cot 23.9 )

If you punch that into your calculator you will solve the problem and get a height of y = 993 ft.

However, this is NOT the height of Uluru (Ayers Rock) according to the Australian Government and other sources which list it as 348 meters (1142 ft) high.

Perhaps Chun-Wei needs to recalibrate his scope!

Cheers,
Stephen La Rocque.

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.