Question

A company plans to manufacture a rectangular box with a square base, an open top, and a volume of 452 cm3. The cost of the material for the base is 0.4 cents per square centimeter, and the cost of the material for the sides is 0.6 cents per square centimeter. Determine the dimensions of the box that will minimize the cost of manufacturing it. What is the minimum cost?

Step 1

The given box is in rectangular shape but has a square base with volume of the box as 452 cm^{3}.

So consider the base and the sides of the rectangular box as x and y, respectively.

Thus, *x*^{2}*y* = 452 cm^{3}.

The cost for base is 0.4 cents and for sides is 0.6.

The surface area for the box can get the cost of the box.

Thus, *c* (*x*, *y*) = 0.4*x*^{2} + 0.6 (4*xy*).

Rewrite the volume as,

Step 2

Substitute *y* in *c* (*x*, *y*) as follows.

Step 3

Now find the value of *x* ...

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