Final answer:
Using the given dimensions and proportional reasoning with similar triangles, the correct answer is that the nuclear power plant's cooling tower is 216 meters tall.
Step-by-step explanation:
The student has asked for help in determining the height of a nuclear power plant's cooling tower which has a hyperbolic cross-section. To calculate the height of the tower, we can use the dimensions given: the base diameter is 198 meters, the diameter at the narrowest point (72 meters above the ground) is 66 meters, and the diameter at the top is 132 meters. Since we are given three points on the hyperbola (the base, the narrowest point, and the top), we can identify the shape of the hyperbola and use similar triangles to determine the height of the tower.
Considering the shape of the tower, the narrowest point divides the hyperbola into two similar sections. The ratio of the base's diameter to the diameter at the narrowest point is 198/66, which equals 3. This ratio will be the same for the two sections of the hyperbola formed by the tower. Given that the height at the narrowest point is 72 meters, we can set up a proportion for the upper section of the tower: as 66 is to 132 (the ratio of the diameters at the narrowest and the top point), so is 72 to the unknown height (H) of the upper section of the tower. Solving this proportion gives us H = 2 × 72 = 144 meters. The total height of the tower is the sum of the heights of both sections: 72 + 144 = 216 meters. Therefore, the correct answer is that the nuclear power plant's cooling tower is 216 meters tall. This method of problem-solving demonstrates the use of geometry and proportional reasoning to solve real-world problems.