trignometry related question

Question

At point the angle of elevation of the top of a tower is such that its tangent is 5/12 . On walking 80m towards the tower, the cotangent of the angle of elevation of the top of the tower is 4/3. The height of the tower is how much m?

Answer 1

Survey formula:

$$\frac{S(\tan a_1.\tan a_2)}{(\tan a_1-\tan a_2)}$$

$$\frac{80(3/4)(5/12)}{(\frac34-\frac5{12})}= 75 \space\text{ m}$$

Answer 2

Using the information given, you can derive the following two equations: $$ (i):\; \frac{h}{x} = \frac{5}{12}\\ (ii):\;\frac{h}{x-80} = \frac{3}{4} $$

Solving (i) for x, (ii) for h, and combining both gives $$ (i):\; x = h\frac{12}{5}\\ (ii):\; h = (x-80)\frac{3}{4}\\ (i+ii):\; h = (h\frac{12}{5}-80)\frac{3}{4}\\ h = h\frac{9}{5}-60\\ h = \frac{60}{\frac{9}{5}-1} = 300/4 = 75 $$

So the height is 75m.