The dimensions that will minimize the cost are:
Radius (r): 7.81 ft (rounded to three decimal places)
Height (h): 11.26 ft (rounded to three decimal places)
Solving for the dimensions
Let:
r be the radius of the cylinder and the hemispheres
h be the height of the cylinder
Step 1: Find the volume equations
The volume of the two hemispheres is:
V_hemispheres = 2 * (2/3)πr^3
The volume of the cylinder is:
V_cylinder = πr^2h
The total volume of the tank must be 4480 cubic feet:
V_total = V_hemispheres + V_cylinder = 4480
Substitute the volume equations and simplify:
2 * (2/3)πr^3 + πr^2h = 4480
Step 2: Find the surface area equations
The surface area of the two hemispheres is:
S_hemispheres = 2 * 4πr^2
The surface area of the cylinder is:
S_cylinder = 2πrh + 2πr^2
The total surface area of the tank is to be minimized:
S_total = S_hemispheres + S_cylinder
Substitute the surface area equations:
S_total = 2 * 4πr^2 + 2πrh + 2πr^2
Step 3: Factor and rewrite the cost function
Since the hemispherical ends cost twice as much per square foot of surface area as the sides, we can represent the cost as:
C = k * (2 * 4πr^2 + 2πrh) + S_cylinder
where k is a cost constant.
Factor out 2πr:
C = k * 2πr(4r + h) + S_cylinder
Substitute the volume equation for h:
C = k * 2πr(4r + (4480 - 2πr^3) / πr^2) + S_cylinder
Simplify:
C = k * (8πr^2 + 8960 / r - 4πr) + S_cylinder
Step 4: Minimize the cost function
To minimize the cost, we need to find the minimum value of the function. We can do this by taking the derivative of the cost function with respect to r and setting it to zero:
dC/dr = k * (16πr - 8960 / r^2 - 4π) = 0
Solve for r:
r^3 = 560
r ≈ 7.81 ft
Substitute the value of r back into the volume equation to find h:
h = (4480 - 2πr^3) / πr^2 ≈ 11.26 ft
Step 5: Round the answers
Therefore, the dimensions that will minimize the cost are:
Radius (r): 7.81 ft (rounded to three decimal places)
Height (h): 11.26 ft (rounded to three decimal places)