>> A radio tower is located at 425 feet from a building.
From a window in the building a person determines that the angle of elevation
to the top of the tower is 38 degrees and that the angle of depression
to the bottom of the tower is 30 degrees.
How tall is the tower?
So to set this up, this blue rectangle will be the building with a person who's looking
out the window at this location here.
This red rectangle represents the tower.
And because the tower is located at 425 feet from the building,
we know that this length here is 425 feet.
This would also be the person's line of sight.
Because the angle of elevation to the top
of the tower is 38 degrees this angle here would be 38 degrees.
And the angle of depression to the bottom of the tower is 30 degrees,
and therefore this angle here would be 30 degrees.
Notice how we have two right triangles here, one above the line
of sight and one below the line of sight.
Our goal is to determine the height of the tower, which would be this length here.
So looking at this lower right triangle, if we label the length of this leg here Y sub 1,
then looking at the upper right triangle, if we label this leg Y sub 2, notice how the height
of the tower would be equal to Y sub 1 plus Y sub 2.
And because we have two right triangles here we can now use right triangle trigonometry
to determine the length of Y sub 1 and Y sub 2.
So first, looking at the lower right triangle notice [inaudible] we know this angle here,
Y sub 1, would be the opposite side, and we also know the length
of this leg, the adjacent side is 425 feet.
So because we'll be using this angle on the opposite side
and the adjacent side we should recognize that we'll be using the tangent function
to determine the length of Y sub 1.
Looking at the upper right triangle, again, if we use this angle,
notice how Y sub 2 is the opposite side.
And we also know the adjacent side has a length of 425 feet.
So in both cases we'll be using the tangent function to determine Y sub 1 and Y sub 2.
So again, looking at the lower right triangle we can say that the tangent
of 30 degrees must be equal to the ratio of the opposite side
to the adjacent side or Y sub 1 divided by 425 feet.
[ Demonstration ]
Then looking at the upper right triangle we can say that the tangent of 38 degrees is equal to,
again, the opposite side divided by the adjacent side or Y sub 2 divided by 425 feet.
Notice how in both cases, to solve for Y sub 1 and Y sub 2,
we would multiply both sides of the equation by 425.
Looking at the right side notice how this would simplify to 1 and so would this; therefore,
Y sub 1 is equal to 425 times tangent 30 degrees
and Y sub 2 is equal to 425 times tangent 38 degrees.
And again, because the height is equal to Y sub 1 plus Y sub 2 we can say the height is equal
to 425 tangent 30 degrees plus 425 tangent 38 degrees.
And now we go to the calculator to get an approximate height.
The first thing we should do is check to make sure the calculator is in degree mode
because our angles are in degrees.
So we'll press the mode key, go down to the third row, highlight degree, press enter,
go back to the home screen by pressing second mode for quit
and then we'll enter 425 tangent 30 plus 425 tangent 38 degrees.
Enter. If we rounded to two decimal places,
notice how the approximate height would be 577.42 feet.
[ Demonstration ]
I hope you found this helpful.
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