Question

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.

Answer 1

Answer:

23.32 degrees

Step-by-step explanation:

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.  

$sin62=\frac{x}{95}$ 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 $tan62=\frac{83.88}{y}$.  Now let's do the same thing for the other triangle with the unknown angle.

$tan\beta =\frac{83.88}{y+150}$

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

$y=\frac{83.88}{tan62}$

The second one solved for y will simplify to:

$y=\frac{83.88-150tan\beta }{tan\beta }$

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:

$\frac{83.88-150tan\beta }{tan\beta }=\frac{83.88}{tan62}$

Cross multiply to get this big long messy looking thing:

$tan62(83.88-150tan\beta )=83.88tan\beta$

Distribute through the parenthesis to get

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

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

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

And then factoring it out gives you:

$83.88tan62=tan\beta(83.88+150tan62)$

Divide to get

$tan\beta =\frac{83.88tan62}{83.88+150tan62}$

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!