A cooling tower for a nuclear reactor is to be constructed in the shape of a hyperboloid of one sheet. The diameter at the base is 260 m and the minimum diameter, 500 m above the base, is 200 m. Find an equation for the tower. (Assume the center is at the origin with axis the z-axis and the minimum diameter is at the center.)
To find an equation for the tower in the shape of a hyperboloid of one sheet, we can use the standard equation for a hyperboloid of one sheet with its axis aligned with the z-axis:
(x^2 / a^2) - (y^2 / b^2) - (z^2 / c^2) = 1
In order to determine the values of a, b, and c, we need to consider the given information about the tower.
1. The diameter at the base is 260 m, which means the radius at the base is 130 m. Since the base is in the xy-plane, this gives us a^2 + b^2 = 130^2.
2. The minimum diameter, 500 m above the base, is 200 m. This means the radius at this point is 100 m. Since this point is (0, 0, 500), we have c^2 = 500^2 + 100^2.
Now we have two equations:
a^2 + b^2 = 130^2
c^2 = 500^2 + 100^2
Let's solve these equations to find the values of a, b, and c.
First, substitute a^2 from the first equation into the second equation:
(130^2 - b^2) + b^2 = 500^2 + 100^2
Simplifying this equation:
16900 - b^2 + b^2 = 250000 + 10000
16900 = 260000
b^2 = 260000 - 16900
b^2 = 243100
Taking the square root of both sides:
b = √243100
Now, substitute the value of b into the first equation:
a^2 + (√243100)^2 = 130^2
a^2 + 243100 = 16900
a^2 = 16900 - 243100
a^2 = 226200
Taking the square root of both sides:
a = √226200
Finally, we substitute the values of a, b, and c into the equation of the hyperboloid:
(x^2 / (√226200)^2) - (y^2 / (√243100)^2) - (z^2 / (√(500^2 + 100^2))^2) = 1
Simplifying further:
x^2 / 226200 - y^2 / 243100 - z^2 / 251000 = 1
Therefore, the equation for the tower in the shape of a hyperboloid of one sheet is:
x^2 / 226200 - y^2 / 243100 - z^2 / 251000 = 1