Question

A guy wire to a tower makes a 72 degrees angle with level ground. At a point 30 ft farther from the tower than the wire but on the same side as the base of teh wire, the angle of elevation to the top of the tower is 30 degrees. Find the length of the wire (to the nearest foot).

Answer 1

so.. checking the picture below

we can say y = y, or just do a substitution and end up with


`\bf tan(72^o)=tan(30^o)(30+x)\implies 3.08x=\cfrac{30}{√(3)}+\cfrac{1}{√(3)}x \\\\\\ 3.08x-\cfrac{1}{√(3)}x=\cfrac{30}{√(3)}\implies 2.5x\approx 17.32\implies x\approx \cfrac{17.32}{2.5}`

once you know, how long is "x", then you can simply use the cosine of 72° to get "r"

thus
`\bf cos(\theta)=\cfrac{adjacent}{hypotenuse}\qquad cos(72^o)=\cfrac{x}{r}\implies r=\cfrac{x}{cos(72^o)}`

Answer 2

`r=21.45{\text{feet}`

Explanation:

To find: The length of the wire.

Solution:

From the figure, using trigonometry, we have

`tan72^(\circ)=(y)/(x)`

⇒
`y=xtan72^(\circ)` (1)

And,
`tan30^(\circ)=(y)/(30+x)`

⇒
`y=(30+x)tan30^(\circ)` (2)

Thus, from equation (1) and (2), we get

`xtan72^(\circ)=(30+x)tan30^(\circ)`

⇒
`(x)/(30+x)=(tan30^(\circ))/(tan72^(\circ))`

⇒
`(x)/(30+x)=(0.577)/(3.077)`

⇒
`(x)/(30+x)=0.181`

⇒
`x=5.43+0.181x`

⇒
`x=6.63{\text{feet}`

Also,
`(x)/(r)=cos72^(\circ)`

⇒
`(x)/(cos72^(\circ))=r`

⇒
`(6.63)/(0.309)=r`

⇒
`r=21.45{\text{feet}`

Therefore, the length of the wire is 21.45 feet.