The Problem:

The spherical top holds a little over 54,000 gallons of water. The base of the top is 60 feet above the ground. The diameter of the supporting column is one-third the radius of the sphere. The base of the supporting column flares out like a cone (a frustum) so that the diameter of the bottom is equal to the radius of the sphere; this bottom section is 15 feet tall. Likewise, the connection between the support and the sphere (another frustum) flares out to a diameter equal to two-thirds the radius of the spehere and is 10 feet tall.
Paint must be applied at an average thickness of .16cm over the entire structure. Calculate how much pain is needed to paint the details (Surface Area of total structure).

Is this math or english

To calculate the surface area of the entire structure and determine the amount of paint needed, we need to calculate the surface area of each component separately and then sum them up.

1. Spherical Top:
The formula for the surface area of a sphere is given by:
A_s = 4πr^2

Given that the top holds a little over 54,000 gallons of water, we can convert it to cubic feet and use it to calculate the radius of the sphere.

Volume of water = 54,000 gallons
1 gallon = 231 cubic inches
1 cubic foot = 12^3 = 1,728 cubic inches
Volume of water = 54,000 * 231 / 1,728 cubic feet

Knowing the volume of a sphere is given by:
V_s = (4/3)πr^3

We can solve for the radius (r) using the volume of the water in the sphere and then calculate its surface area.

2. Supporting Column Cone (Bottom Section):
The formula for the surface area of a cone (frustum) is given by:
A_c = π(R + r)s

where R is the radius of the larger base, r is the radius of the smaller base, and s is the slant height.

Given that the diameter of the bottom section is equal to the radius of the sphere, and the bottom section is 15 feet tall, we can calculate its surface area using the formula above.

3. Connection between Support and Sphere (Another Frustum):
Similar to the supporting column cone, we can calculate the surface area of the frustum connecting the support and the sphere using the same formula, given that the diameter of the bottom section is equal to two-thirds the radius of the sphere and the height is 10 feet.

Once we have calculated the surface area of each component, we can sum them up to get the total surface area of the entire structure. Finally, we can multiply the surface area by the average thickness of the paint (.16cm) to determine the amount of paint needed.