You are in charge of a tank to be used for water storage. Your boss has told you that he wants a cylindrical tank with a minimum capacity of 300,000 cubic feet. Your boss wants to spend the least possible amount in painting the tank. Design a tank to meet this objective.
Let y be the height. The diameter D must satisfy
(pi/4) D^2 y = 3*10^5 ft^3
The area to be painted (including the top) is
A = pi [D^2/4 + D y]
Substitute using the first equation to eliminate y.
A = pi [D^2/4 + 3*10^5 (4/pi)(1/D)]
Set dA/dD = 0 and solve for D
D/2 - (12*10^5/pi)/D^2 = 0
D^3 = 24*10^5/pi
D = 92.6 ft
To design a tank that meets the objective of having the least possible amount spent on painting, there are a few key factors to consider: the shape of the tank, the material used, and the surface area of the tank.
First, let's consider the shape of the tank. A cylindrical tank is a good choice because it maximizes volume while minimizing surface area. This means we can store the required capacity of 300,000 cubic feet while minimizing the amount of paint needed to cover the tank's surface.
Next, let's consider the material of the tank. Choosing a material that does not require painting can help reduce costs significantly. One such material is stainless steel. Stainless steel tanks are commonly used for water storage because they have corrosion-resistant properties, reducing the need for painting.
Finally, let's consider the surface area of the tank. The amount of paint required will depend on the surface area that needs to be covered. A key factor in minimizing surface area is the height-to-diameter ratio of the tank. By choosing a tank with a greater height-to-diameter ratio, you can reduce the surface area exposed to the outside and therefore minimize the amount of paint needed.
To summarize, design a cylindrical tank made of stainless steel with a greater height-to-diameter ratio. This will help maximize the tank's capacity while reducing the surface area that needs to be painted, ultimately minimizing the cost of painting.