Final answer:
By establishing a ratio based on similar triangles formed by the dimensions at different heights of the cooling tower, the width of the tower at 210 feet can be calculated using the given dimensions.
Step-by-step explanation:
The student's question involves using the properties of a hyperbola to find the width of a nuclear power plant cooling tower at a certain height. Given that the smallest diameter of the tower is 143 feet and occurs at a height of 411.5 feet, while the tower's diameter at the base is 253 feet, we can use similar triangles to determine the width at a height of 210 feet from the ground.
First, we know the shape of the cooling tower is like a hyperbola, and by using the information given, we can find out the dimensions of the tower at different heights by understanding that the ratios of corresponding sides of similar triangles are equal. This establishes the proportions we need to find our unknown width.
Let's call the width of the tower at 210 feet from the ground 'w'. To find 'w', we set up a proportion based on the given information:
The change in width from the smallest diameter to the base diameter is 253 feet - 143 feet:
Change in width = 253 - 143 = 110 feet
The change in height from the smallest diameter location to the base is:
Total height - Height at smallest diameter = 587 - 411.5 = 175.5 feet
Now, we create a ratio using the given height of 210 feet:
Ratio = (210 - 411.5) / 175.5
Using this ratio, we can find 'w' by multiplying the ratio by the total change in width:
w = (210 - 411.5) / 175.5 × 110 feet
Then, we can add this value to the smallest diameter to get the final width:
Width at 210 feet = 143 + w
By performing the calculation, you'll find the width of the tower at 210 feet above the ground.