Applications of Cubic Functions Volume of a Open

Applications of Cubic Functions

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 x x x 16 16 - 2 x 20 - 2 x x

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

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

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

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

What size square should be cut from each corner to realize the maximum volume? • What do you want now? • X!! • 2. 9 inches

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

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