The Height of a Tower

  • From: Davinda
  • Date: 30 August 1999
  • Subject: Strange Trigonometry Question

We have done some questions like this before, but I can't do this one
because they don't tell you the distance along the ground to the tower.

Jackie is walking towards a tower. At one point she measures the
angle of elevation to be 23 degrees. When she has walked another 50m,
the angle of elevation is 40 degrees.
What is the height of the tower?

Maths Help suggests:

Diagram of measurements

The first thing to do is a sketch diagram showing all the measurements you are given in the question.

The height of the tower we have called h, and
the distance from the foot of the tower to the nearer measurement is x.

Looking at the red triangle, tan40=h/x

and similarly for the blue triangle, tan23=h/(50+x)

We must use these two equations to eliminate x and to find the value of h.

Rearranging the first equation to make x the subject gives x=h/tan40

and the second equation gives 50+x=h/tan23

Now substitute for x into the second equation and rearrange:
h = 50/(cot23-cot40)

The height of the tower works out to be 42.95 metres (to 4 significant figures).

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