SOLUTION: the cross-section of a nuclear power plants cooling tower is in the shape of a hyperbola. suppose the tower has a base diameter of 222 meters and the diameter at its narrowest poin

Question 1058490: the cross-section of a nuclear power plants cooling tower is in the shape of a hyperbola. suppose the tower has a base diameter of 222 meters and the diameter at its narrowest point, 72 meters above the ground, is 74 meters. if the diameter at the top of the tower is 148 meters, how tall is the tower?
This is not part of a test but I could not find the answere to it.
Answer by solve_for_x(190) (Show Source): You can put this solution on YOUR website!
The general equation of a hyperbola is:

where (h, k) is the coordinate pair of the center of the hyperbola.

Assuming that the y-axis of our coordinate system runs vertically
through the center of the tower, the coordinates of the center of
the hyperbola are h = 0, k = 72.

The equation of the hyperbola is then:

From the description given, we know two points on the curve of
the hyperbola.

Since the diameter is 222 meters at ground level, the point x = 111,
y = 0 is on the curve. (You divide the diameter in half because we have
the y-axis of our coordinate system running through the middle of the
tower. See diagram below.)

Also, since the diameter is 74 meters at a height of 72 meters, we know
that the point (37, 72) is also on the curve of the hyperbola.

Substituting x = 37, y = 72 gives:

Updating the equation with a^2 = 1369 then gives:

Substituting the other known point (111, 0) gives:

The equation can again be updated to give:

Now, at the top of the tower, the width is 148 meters,
which corresponds with an x-coordiante of 148/2 = 74.

Substituting 74 in place of x and solving for y gives:

y = 116.09 meters

Solution: The height of the tower is 116.09 meters.

Graph:

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