A 95-foot wire attached from the top of a cell phone tower makes a 62 degree angle with the ground. Joey is standing 150 feet behind the wire, looking up at the tower. Find the angle of elevation from the point on the ground where Joey is standing to the top of the tower.

We set up a large right triangle that has 2 triangles within it. The large triangle is a right triangle. The height of it is the height of the tower, the base angle is 62, the hypotenuse is 95, and the base measure is y. The other triangle has the same height which is the height of the tower, the angle is what we are looking for, and the base measure is 150 feet beyond y, so its measure is y + 150. We have enough information to find the height of the tower, so let's do that first. Going back to the first smaller triangle.

[tex]sin62=\frac{x}{95}[/tex] so the height of the tower is 83.88 feet. Now we need to solve for y. Using that same triangle and the tangent ratio, we find that [tex]tan62=\frac{83.88}{y}[/tex]. Now let's do the same thing for the other triangle with the unknown angle.

[tex]tan\beta =\frac{83.88}{y+150}[/tex]

Solve both of these for y. The first one solved for y:

[tex]y=\frac{83.88}{tan62}[/tex]

The second one solved for y will simplify to:

[tex]y=\frac{83.88-150tan\beta }{tan\beta }[/tex]

Now that these are both solved for y, and y = y, we can set them equal to each other by the transitive property of equality:

[tex]\frac{83.88-150tan\beta }{tan\beta }=\frac{83.88}{tan62}[/tex]

Cross multiply to get this big long messy looking thing:

[tex]tan62(83.88-150tan\beta )=83.88tan\beta[/tex]

Distribute through the parenthesis to get

[tex]83.88tan62-[(tan62)(150tan\beta)]=83.88tan\beta[/tex]

Get the unknown angles on the same side so it can be factored out:

[tex]83.88tan62=83.88tan\beta +[(tan62)(150tan\beta )][/tex]

And then factoring it out gives you:

[tex]83.88tan62=tan\beta(83.88+150tan62)[/tex]

Divide to get

[tex]tan\beta =\frac{83.88tan62}{83.88+150tan62}[/tex]

Do this on your calculator in degree mode to give you an angle measure of 23.32°. I know this is really hard to follow without being able to draw the pics for you like I do in my classroom, but hopefully you can follow my description and draw your own triangles and follow from that!

oeerivona oeerivona · Answer 2 · 2021-10-14T12:54:15+02:00

An angle of elevation is formed by two reference lines; the horizontal line and the line from the reference point to the point in view

The angle of elevation from the point on the ground where Joey is standing to the top of the tower is approximately 23.318°

Reason:

Given parameter;

Length of the wire = 95 ft.

Angle formed by the wire and the ground = 62°

Location Joey is standing behind the wire, L = 150 feet

Required:

The angle of elevation from the ground at the point where Joey is standing to the top of the tower

Solution:

The height of the tower, h = 95 × sin(62°) ≈ 83.88 feet

Distance of the tower to the point the wire touches the ground, d, is given as follows;

d = 95 × sin(62°) ≈ 44.6 ft.

The distance of of Joey from the base of the tower, D = L + d

D = 150 ft. + 44.6 ft. = 194.6 ft.

Distance of Joey from the base of the tower, D = 194.6 ft.

Let θ represent the angle of elevation from the point on the ground where Joey is standing to the top of the tower, we have;

[tex]tan(\theta ) = \dfrac{h}{D}[/tex]

Therefore;

[tex]\theta = arctan \left(\dfrac{h}{D} \right)[/tex]

Which gives;

[tex]\theta = arctan \left(\dfrac{83.88}{194.6} \right) \approx 23.318^{\circ}[/tex]

The angle of elevation from the point on the ground where Joey is standing to the top of the tower, θ ≈ 23.318°

Learn more about angle of elevation here:

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