Question

A 17 m guy wire attached to the top of a tower (the height of the tower is not yet known) is anchored on the ground, 8 m away from the base of the tower. A second guy wire needs to be attached to the centre of the tower and then anchored to the same ground-anchor as the first wire.Draw and label a diagram.How long does the second guy wire need to be?Determine the measure of the angle formed between the two wires.

Answer 1

Solution

Step 1:

Draw the diagram to illustrate the information.

Step 2:

Use the Pythagoras theorem to find the height of the pole.

$\begin{gathered} 17^2\text{ = h}^2\text{ + 8}^2 \\ 289\text{ = h}^2\text{ + 64} \\ h^2\text{ = 289 - 64} \\ h^2\text{ = 225} \\ h\text{ = }\sqrt{225} \\ \text{h = 15m} \end{gathered}$

Step 3

b) The height of the second guy wire = d

$\begin{gathered} \text{Apply the pythagoras theorem} \\ d^2=\text{ \lparen}\frac{15}{2}\text{\rparen}^2+\text{ 8}^2 \\ d^2\text{ = 56.25 + 64} \\ d^2\text{ = 120.25} \\ \text{d = }\sqrt{120.25} \\ \text{d = 10.97m} \end{gathered}$

c)

$\begin{gathered} sin(\theta\text{ + }\alpha)\text{ = }\frac{Opposite}{Hypotenuse} \\ sin(\theta\text{ + }\alpha)\text{ = }\frac{15}{17} \\ \theta\text{ + }\alpha\text{ = sin}^{-1}(\frac{15}{17}) \\ \theta\text{ + }\alpha\text{ = 61.9}^o \end{gathered}$$\begin{gathered} sin\alpha\text{ = }\frac{7.5}{10.97} \\ \alpha=\text{ sin}^{-1}(\frac{7.5}{10.97}) \\ \alpha\text{ = 43.1} \end{gathered}$

The angle between the two guys' wires = 61.9 - 43.1

Measure of the angle formed between the two wires = 18.8