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Home > PC3 > Chapter 4 > Lesson 4.1.1 > Problem 4-13

4-13.

The manufacturing company you work for has been hired to produce a cubic foot box. It must have a square base and no top. The material for the base costs per square foot and the material for the sides costs per square foot. Complete the parts below to determine the minimum cost of producing the box.   

  1. Sketch a diagram of this situation.

    Sketch a box and label the sides of the base and the height .

  2. Write an expression for the cost of the base in terms of .

    base cost (area of the base)

  3. Express the cost of the sides in terms of and , the height of the box.

    side cost (length of base)(height)
    How many sides does the box have?

  4. Express the total cost of the box in terms of and .

    total cost base cost cost of all sides

  5. Use the fact that the volume of the box is cubic feet to write an equation for the volume of the box.

    (area of the base)(height)
    (area of the base)(height)

  6. Solve for h in your equation from part (e) and then express the cost only in terms of .

  7. Graph your equation from part (f) to determine value of that gives the minimum cost. What is the minimum cost?