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

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{ = }√(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}(15)/(2)\text{\rparen}^2+\text{ 8}^2 \\ d^2\text{ = 56.25 + 64} \\ d^2\text{ = 120.25} \\ \text{d = }√(120.25) \\ \text{d = 10.97m} \end{gathered}`

c)

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