The cooling tower of a nuclear power plant has a hyperbolic cross-section. The tower has a base diameter of 222 meters and the diameter at its narrowest point is 74 metres and is 72 meters above the ground. Given that the diameter at the top of the tower is 148 meters, how tall is the tower?
Question
The cooling tower of a nuclear power plant has a hyperbolic cross-section. The tower has a base diameter of 222 meters and the diameter at its narrowest point is 74 metres and is 72 meters above the ground. Given that the diameter at the top of the tower is 148 meters, how tall is the tower?
Solution
To solve this problem, we need to use the properties of a hyperbola.
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First, we need to find the equation of the hyperbola. We know that the hyperbola is symmetric about the y-axis, so it has the form (x/h)^2 - (y/k)^2 = 1, where h is the horizontal semi-axis and k is the vertical semi-axis.
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We know that the base diameter of the tower is 222 meters, so the length of the horizontal semi-axis h is 222/2 = 111 meters.
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We also know that the diameter at the narrowest point is 74 meters, so the length of the vertical semi-axis k when y = 72 is 74/2 = 37 meters.
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Substituting h = 111 and k = 37 into the equation of the hyperbola, we get (x/111)^2 - (y/37)^2 = 1.
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We also know that the diameter at the top of the tower is 148 meters, so when x = 148/2 = 74, we can solve for y.
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Substituting x = 74 into the equation of the hyperbola, we get (74/111)^2 - (y/37)^2 = 1. Solving for y, we get y = sqrt[(1 - (74/111)^2) * 37^2].
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Calculating the square root, we get y = 111. This is the height of the tower above the narrowest point.
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Adding this to the height of the narrowest point above the ground, we get the total height of the tower is 111 + 72 = 183 meters.
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