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Calculus/minimizing multilpe variable

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Expert: - 4/4/2008


Question
An aquarium is being built in the shape of a rectangular solid. The top is open, the sides are glass, and the bottom is made from a thick plastic. If the cost per square cm of plastic is 1.3 times the cost per square cm of glass and the tank is to hold 670000 cubic cm, what dimensions will minimize the cost?

Answer
Give the tank height h, width w, and length l.  When they are all multiplied together, tht total volume in is gotten.  Note that it is in cm.

Cost 2 times the area of two sides plus 2 times the area of the other two side.

The cost of the base is 1.3 times the base squared.

Total cost is gotten by adding these three factors together.

Note in a rectangle on the bottom, the best way to shape it will be a square, so the width and length are equal.

Let m be money.  The area on the sides is to be taken normally, but the area of the base is to be multiplied by 1.3.

Working with the volume equation, note that the area of the base times the height is the volume, so the height can be solved for in terms of the base.

Put this all together and the volume can be found as a function of the base width and differentiated then set to 0.

Best of luck and more questions are welcome.  Be sure and ask me on what you don't understand yet.

Calculus

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