You are given a rectangular sheet of cardboard that measures 11 in. by 8.5 in. (see the diagram below). A small square of the same size is cut from each corner, and each side folded up along the cuts to from a box with no lid. 1. Anya thinks the cut should be 1.5 inches to create the greatest volume, while Terrence thinks it should be 3 inches. Explain how both students can determine the formula for the volume of the box. Determine which student's suggestion would create the larger volume. Explain how there can be two different volumes when each student starts with the same size cardboard. 2. Why is the value of x limited to 0 in. < x < 4.25 in.?
Answers
Given : rectangular sheet of cardboard that measures 11 in. by 8.5 in A small square of the same size is cut from each corner, and each side folded up along the cuts to from a box with no lid.
To find : Maximum Volume of the box
Solution:
Rectangular Sheet = 11 inch * 8.5 inch
Square of x size is cut from Each corner hence
Remaining Size = 11 - x - x = 11 - 2x inch & 8 .5 - 2x inch
and height = x inch
8.5 > 8 .5 - 2x > 0
=> 0 < x < 4.25
Volume = (11 - 2x)(8.5 - 2x)(x)
Volume = ( 4x² -39x + 93.5)x
=> Volume = 4x³ - 39x² + 93.5x
dV/dx = 12x² - 78x + 93.5
dV/dx = 0
=> 12x² - 78x + 93.5 = 0
=> x = 4.914 , 1.585
x = 4.914 not possible as 0 < x < 4.25
Hence x = 1.585
d²V/dx² = 24x - 78
at x = 1.585
= - 39.9 < 0
Hence Volume is max at x = 1.585 inch
Volume = 66.15 inch³
Volume at x = 1.5 => 8 * 5.5 * 1.5 = 66
Volume at x = 3 => 5 * 2.5 * 3 = 37.5
Anya gets higher Volume
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