Question

A point H is 20m away from the foot of a tower on the same horizontal ground. From the point H, the angle of elevation of the point (P) on the tower and the top(T) of the tower are 30° and 50° respectively.

( a) draw a diagram to illustrate the information above.
(b) calculate correct to 3 s.f,
( I) /PT/
(ii) the distance between H and the too of the tower.
(III) the position of H if the angle of depression of H from the too of the tower is to be 40°

Answer 1

a. See Attachment 1

b.
`PT = 12.3\ m`

c.
`HT = 31.1\ m`

d.
`OH = 28.4\ m`

Explanation:

Calculating PT

To calculate PT, we need to get distance OT and OP

Calculating OT;

We have to consider angle 50, distance OH and distance OT

The relationship between these parameters is;

`tan50 = (OT)/(20)`

Multiply both sides by 20

`20 * tan50 = (OT)/(20) * 20`

`20 * tan50 = OT`

`20 * 1.1918 = OT`

`23.836 = OT`

`OT = 23.836`

Calculating OP;

We have to consider angle 30, distance OH and distance OP

The relationship between these parameters is;

`tan30 = (OP)/(20)`

Multiply both sides by 20

`20 * tan30 = (OP)/(20) * 20`

`20 * tan30 = OP`

`20 * 0.5774= OP`

`11.548 = OP`

`OP = 11.548`

`PT = OT - OP`

`PT = 23.836 - 11.548`

`PT = 12.288`

`PT = 12.3\ m` (Approximated)

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Calculating the distance between H and the top of the tower

This is represented by HT

HT can be calculated using Pythagoras theorem

`HT^2 = OT^2 + OH^2`

Substitute 20 for OH and
`OT = 23.836`

`HT^2 = 20^2 + 23.836^2`

`HT^2 = 400 + 568.154896`

`HT^2 = 968.154896`

Take Square Root of both sides

`HT = √(968.154896)`

`HT = 31.1\ m` (Approximated)

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Calculating the position of H

This is represented by OH

See Attachment 2

We have to consider angle 50, distance OH and distance OT

The relationship between these parameters is;

`tan50 = (OH)/(OT)`

Multiply both sides by OT

`OT * tan50 = (OH)/(OT) * OT`

`OT * tan50 = {OH`

`OT * 1.1918 = OH`

Substitute
`OT = 23.836`

`23.836 * 1.1918 = OH`

`28.4= OH`

`OH = 28.4\ m` (Approximated)