Hello, calc_help123!

A tank with a rectangular base and rectangular sides is to be open at the top.

It is to be constructed so that its width is 4 m and its volume is 36 m³.

If building the tank costs $10/m² for the base and $5/m² for the sides,

what is the cost of the least expensive tank? We can **not** separate the surface area and the cost of materials.

. . We must come up with a single Cost Function ... which we will minimize. Code:

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x

The width of the box is 4 m.

Let = length.

Let = height.

The volume will be 36 m³.

So we have: . .[1]

Now we will create a Cost Function.

The front and back has area: . m².

The left and right has area: . m².

The total area of the sides of the box is: . m².

. . At $5/m², the sides will cost: . dollars.

The base has area: . m².

. . At $10/m², its cost is: . dollars.

Hence, the total cost is: . .[2]

Substitute [1] into [2]: .

. . and we have: .

And **that** is the function we must minimize . . .