solving the triangle

Question

The angle of elevation to the top of a tower from point A on the ground is

15.4°.

From a point​ B,

32.8

feet closer to the​ tower, the angle of elevation is

18.5°.

What is the height of the​ tower?

Answer 1

Let x = distance from point A to the base of the tower and let h = height of the tower

tan15.4° = h / x

So, h = xtan15.4°

tan18.5° = h / (x - 32.8)

So, h = (x - 32.8)tan18.5°

Therefore, xtan15.4° = (x - 32.8)tan18.5°

Solve for x, then solve for h.

Answer 2

Let the distance from the base of the tower to point B be x feet.

Then the distance from the base of the tower to point A is x+32.8 feet.

Make a diagram and see that tan (18.5º) = h/x (Remember that tan is "opposite over adjacent".)

To get the above h all by itself, multiply both sides of the equation by x:

x·(tan 18.5º) = h

The diagram also shows that tan (15.4º) = h/(x + 32.8) (One again, "opposite over adjacent".)

To get the above h all by itself, multiply both sides of this equation by (x + 32.8):

(x + 32.8)(tan 15.4º) = h

We just produced two different ways of writing the same number, h, so they must be equal:

x·(tan 18.5º) = h = (x + 32.8)tan 15.4º

Let's solve for x. First, use the distributive law:

x·(tan 18.5º) = x·(tan 15.4º) + 32.8·(tan 15.4º)

Now subtract x·tan (15.4º) from both side to get all the x's on the same side (the right side) of the equation:

x·(tan 18.5º)- x·(tan 15.4º) = 32.8·tan(15.4º)

Use the distribute law so that there will be only one x:

x ((tan 18.5º) - tan (15.4º)) = 32.8·tan(15.4º)

Divide both sides by (tan 18.5º) - tan (15.4º) to get the x all by itself (on the left side of the equation):

x = 32.8·tan(15.4º) / ((tan 18.5º) - tan (15.4º))

Now look up these tangents and compute the value of x.

But we're not done yet. The problem asks for h, the height of the tower.

We can get h very simply from this equation, which we was earlier in this solution:

x·(tan 18.5º) = h

Now that we know what x is, multiply it by (tan 18.5º) to get h. Good luck.