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Applications of Cubic Functions

This application of cubic functions explores the construction of an open-top box from a 20-inch by 16-inch cardboard sheet. By cutting squares from each corner, we derive a volume function in terms of the cut size, ( x ). The volume formula, ( V = (20 - 2x)(16 - 2x)x ), guides us to analyze maximum volume and domain constraints. The optimal cut size for maximum volume is 2.9 inches, yielding 420 cubic inches. Further calculations allow for determining volumes at different cut sizes, enhancing our understanding of cubic functions in real-life applications.

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Applications of Cubic Functions

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  1. Applications of Cubic Functions

  2. x x x x x x x x Volume of a Open Box. Suppose you are trying to make an open-top box out of a piece of cardboard that is 20 inches by 16 inches. You are to cut the same size square from each corner. Write a function to represent the volume of this box. 20 20 - 2x 16 - 2x 16

  3. V=lwh 20 - 2x ? x 16 - 2x 20 - 2x

  4. Formula for the Volume of a Box The final answer for the volume will ALWAYS have the term :

  5. 16 -2x 20 -2x Write the formula for the volume of our box: Step 1: Multiply the two binomials together Step 2: Multiply by x 320 -40x -32x 4x2

  6. What is the maximum volume? • What is the possible domain for this box? What is the greatest possible value that we can cut out for x? • 0 < X < 8 (Half of the length of the smallest side) • SO, Xmin = 0 and Xmax = 8; ZOOM 0 • Do you want x or y? • Y!!! • 420 cubic inches

  7. What size square should be cut from each corner to realize the maximum volume? • What do you want now? • X!! • 2.9 inches

  8. What size square should you cut from each corner to realize a volume of 300 cubic inches? • What do you know: x or y? • Y!! Let y = 300; find the intersection • 1.3 inches or 5 inches

  9. What is the volume if a square with side 2 inches is cut from each corner? • What do you know; x or y? • X!!! • Go to table; let x = 2 • 384 Cubic inches

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