A cylindrical oil-storage tank is to be constructed for which the following costs apply:

cost per square meter metal for ides $30.00
combined costs of concrete base and metal bottom $37.50(cost per square meter)

top 7.50 (cost per square meter)

The tank is to be constructed with dimensions such that the cost is minimum for whatever capacity is selected.
a) One possible approach to selecting the capacity is to build the tank large enough for an additional cubic meter of capacity to cost $8.(note that this does not mean $8 per cubic meter average for the entire tank.) what is the optimal diameter and optimal height of the tank?
b) instead of the approach used in part(a), the tank is to be of such a size that the cost will be $9 per cubic meter average for the entire storage capacity of the tank. set up the lagrange multiplier equations and verify that they are satisfied by an optimal diameter of 20 m and optimal height of 15m.

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To find the optimal diameter and height of the cylindrical oil-storage tank, we need to minimize the cost based on different approaches described in parts (a) and (b).

a) Approach 1: Adding an Additional Cubic Meter
In this approach, we want to build the tank large enough so that the cost of an additional cubic meter is $8.

To find the optimal diameter and height, we can use the following steps:

Step 1: Set up the cost equation.
Let:
D = Diameter of the tank
H = Height of the tank

The cost equation is given by:
Cost = (2 * π * (D/2) * H * $30) + (π * (D/2)^2 * $37.5) + (π * (D/2)^2 * $7.5)

Step 2: Define the additional cubic meter cost.
Let:
C = Additional cubic meter cost

Using the given information, C = $8.

Step 3: Set up the objective function.
We want to minimize the cost, so our objective function is:
Objective Function = Cost + C

Step 4: Optimize the objective function.
Differentiate the objective function with respect to D and H, and set the derivatives equal to zero to find optimal values.

b) Approach 2: Average Cost Per Cubic Meter
In this approach, we want the tank size to be such that the cost will be $9 per cubic meter average for the entire tank's storage capacity.

To set up the Lagrange multiplier equations and verify the optimal diameter and height, we can use the following steps:

Step 1: Set up the cost equation (same as in approach 1).

Step 2: Set up the constraint equation.
Let:
V = Volume of the tank

The constraint equation is:
Constraint Equation = Cost / V = $9

Step 3: Set up the Lagrange function.
Lagrangian Function = Cost - λ * (Constraint Equation)

Step 4: Optimize the Lagrange function.
Differentiate the Lagrange function with respect to D, H, and λ, and set the derivatives equal to zero to find optimal values. Verify that these optimal values satisfy the Lagrange multiplier equations.

By following these steps, we can find the optimal diameter and height of the cylindrical oil-storage tank for both approaches.