A cross section of a nuclear cooling tower is a hyperbola with equation x^2/90^2-y^2/130^2=1. The tower is 450ft tall and the distance from the top of the tower to the center of the hyperbola is half the distance from the base of the tower to the center of the hyperbola. Find the diameter of the top and the base of the tower.

Show your work please.

the diameter of the top of the tower=274.833ft

the diameter of the base of the tower=452.413ft

To find the diameter of the top and base of the tower, we need to determine the coordinates of the vertices of the hyperbola.

The equation of the hyperbola is x^2/90^2 - y^2/130^2 = 1. Let's refer to the center of the hyperbola as (h, k).

From the given information, we know that the height of the tower is 450ft. Thus, the coordinates of the top of the tower are (h, k + 450/2) = (h, k + 225).

We also know that the distance from the top of the tower to the center of the hyperbola is half the distance from the base of the tower to the center of the hyperbola. So, the coordinates of the base of the tower are (h, k - (450/2)/2) = (h, k - 112.5).

Substituting these coordinates into the equation of the hyperbola, we get two equations:

For the top of the tower:
(h - h)^2/90^2 - (k + 225 - k)^2/130^2 = 1
0/90^2 - 225^2/130^2 = 1
-225^2/130^2 = 1

For the base of the tower:
(h - h)^2/90^2 - (k - 112.5 - k)^2/130^2 = 1
0/90^2 - 112.5^2/130^2 = 1
-112.5^2/130^2 = 1

Simplifying both equations:

-112.5^2/130^2 = 1
-112.5^2 = 130^2
h^2 - k^2 = 130^2 - 90^2
(k + 225)(k - 112.5) = 130^2 - 90^2

156.25k - 50625 = 16900 - 8100
156.25k - 50625 = 8800
156.25k = 59425
k = 59425/156.25
k = 380

Substituting the value of k back into one of the equations:

h^2 - k^2 = 130^2 - 90^2
h^2 - 380^2 = 130^2 - 90^2
h^2 - 144400 = 16900 - 8100
h^2 - 144400 = 87900
h^2 = 232300
h = √232300
h ≈ 481.06

Therefore, the coordinates of the center of the hyperbola are approximately (481.06, 380).

The diameter of the top of the tower is 2 * the x-value of the vertices, which is 2 * 90 = 180 ft.
The diameter of the base of the tower is 2 * the y-value of the vertices, which is 2 * 130 = 260 ft.

Thus, the diameter of the top of the tower is 180 ft and the diameter of the base of the tower is 260 ft.

To start, let's analyze the given equation of the hyperbola:

x^2/90^2 - y^2/130^2 = 1

This equation represents a hyperbola with its center at the origin (0,0), and the semi-major axis being along the x-axis, while the semi-minor axis is along the y-axis. The semi-major axis is represented by 'a' and the semi-minor axis is represented by 'b'.

Now, let's analyze the information about the tower given in the problem:

- The height of the tower is 450 ft.
- The distance from the top of the tower to the center of the hyperbola is half the distance from the base of the tower to the center of the hyperbola.

Now, let's find the values of 'a' and 'b' for the hyperbola equation:

Comparing the equation to the standard form of a hyperbola: (x^2/a^2) - (y^2/b^2) = 1
We have:
a^2 = 90^2
b^2 = 130^2

Now, let's use the given information about the tower to find the distance from the base to the center of the hyperbola:

Let 'h' be the distance from the base to the center of the hyperbola.
According to the problem, the distance from the top to the center is half the distance from the base to the center. So, the distance from the top to the center is h/2.

The total height of the tower is 450 ft, therefore:
h + h/2 = 450
Multiplying through by 2:
2h + h = 900
3h = 900
h = 900/3
h = 300 ft

Now we have the distance from the base to the center of the hyperbola which is 300 ft.

To find the diameter of the top and the base of the tower, we need to calculate the difference between the x-coordinates of the vertices.

The vertices of the hyperbola are located at (+/- a, 0).
Given: a = 90

The x-coordinate of the top vertex is -90, and the x-coordinate of the base vertex is +90.

To find the diameter, we subtract the x-coordinate of the top vertex from the x-coordinate of the base vertex:

Diameter = (90) - (-90)
Diameter = 90 + 90
Diameter = 180 ft

Therefore, the diameter of the top and base of the tower is 180 ft.