Question

What is the height of the tower ?

Hint:

This type of problem, with a central height and two angles on opposite sides, has a general solution form. If the total distance is d and angles are A and B, the height is h = (d)/( A + B). Memorizing this can be a shortcut.

Answer 1

Using the setup from the sketch, we need to find a mathematical expression for the height of the tower, h.

We will use the tangent ratio, (θ) = OppositeAdjacent, in both right-angled triangles. Let the height be h = TB, and let the base segment BP = x. Then the other base segment is BQ = 40 - x.

In right-angled TBP:
(70^) = (TB)/(BP) = (h)/(x) x = (h)/((70^)) = h (70^) --- (1) In right-angled TBQ:
(48^) = (TB)/(BQ) = (h)/(40-x) 40-x = (h)/((48^)) = h (48^) --- (2) Substitute x from equation (1) into the expression 40-x:
BQ = 40 - h (70^) From equation (2), we know BQ = h (48^). Therefore:
h (48^) = 40 - h (70^) Now, solve for h. Move all terms with h to one side:
h (48^) + h (70^) = 40 Factor out h:
h ((48^) + (70^)) = 40 Isolate h:
h = (40)/((48^) + (70^)) This is the exact expression for the height. A numerical value would require a calculator.

The height of the tower is h = (40)/((70^) + (48^)) metres.