Question

An engineer determines that the angle of elevation from her position to the top of a tower is 57o. She measures the angle of elevation again from a point 43 meters farther from the tower and finds it to be 27o. Both positions are due east of the tower. Find the height of the tower.

Answer 1

Answer:

54.65m

Step-by-step explanation:

This is going to be extremely difficult to explain.  We have one large right triangle split up into 2 triangles:  the first one has a base angle of 57 with unknown base length (y) and unknown height (x), and the second one has a base angle of 27 with unknown base length of y + 43 and unknown height (x.  This is the same x from the first one and is what we are looking for...the height of the tower).  We can find the vertex angle of the first triangle because 180 - 90 - 57 = 33.  The side across from the 33 is y and the side adjacent to it is x so we have that tan33 = y/x.  Not enough yet to do anything with.  Our goal is to solve for that y value in order to sub it in to find x.  Next we have to use some geometry.  The larger triangle has a base angle of 27.  The angle within that triangle that is supplementary to the 57 degree angle is 180 - 57 = 123.  So now we have a triangle with 2 base angles measuring 123 and 27, and the vertex angle then is 180 - 123 - 27 = 30.  That vertex angle of 30 added to the vertex angle of the first triangle is 63 degrees total.  Now we can say that tan63 = (y+43)/x.  Now we have 2 equations with 2 unknowns that allows us to solve them simultaneously.  Solve each one for x.  If

$tan33=\frac{y}{x}$, then

$x=\frac{y}{tan33}$.

If

$tan63=\frac{y+43}{x}$, then

$x=\frac{y+43}{tan63}$

Now that these both equal x and x = x, we can set them equal to each other and solve for y:

$\frac{y}{tan33}=\frac{y+43}{tan63}$

Cross multiply to get

$tan33(y+43)=ytan63$

Distribute through the parenthesis to get

y tan33 + 43 tan33 = y tan 63.

Now get the terms with the y in them on the same side and factor out the common y:

y(tan33 - tan63) = -43 tan33

Divide to get the following expression:

$y=\frac{-43tan33}{(tan33-tan63)}$

This division gives you the fact that y = 64.264 m.  Now we add that to 43 to get the length of the large right triangle as 107.26444 m.  What we now is enough information to solve for the height of the tower:

$tan27=\frac{x}{107.2644}$

and x = 56.65 m

Phew!!!!! Hope I didn't lose you too too badly!  This is not an easy problem to explain without being able to draw the picture like I do in my classroom!