a. Squares with sides of length x are cut out of each corner of a rectangular pi
ID: 2893885 • Letter: A
Question
a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 13 ft by 8 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the argest box that can be formed in this way. b. Suppose that in part (a) the original piece of cardboard is a square with sides of length s. Find the volume of the largest box that can be formed in this way. c. Suppose that in part (a) the original piece of cardboard is a rectangle with sides of length L and W. Holding L fixed, find the size of the corner squares x that maximizes the volume of the box as W->oo. a. a. To find the objective function, express the volume V of the box in terms of x. Type an expression.) The interval of interest of the objective function is Simplify your answer. Type your answer in interval notation. Use integers or decimals for any numbers in the expression.) The maximum volume of the box is approximately ft Round to the nearest hundredth as needed.) b. To find the objective function, express the volume V of the box in terms of s and x. (Type an expression.) The maximum volume of the box is (Type an expression using s as the variable.)Explanation / Answer
From the given question,
length of rectangle=13 ft
breadth of rectangle=8 ft
square with side x are cut out from corners.
new length=13-2x
new breadth=8-2x
height=x
volume of cuboid=(13-2x)(8-2x)(x)
V=(13-2x)(8x-2x2)
=104x-26x2-16x2+4x3
=4x3-42x2+104x
constraints:
13-2x>0 or 8-2x>0 or x>0
x<6.5or x<4 or x>0
so 0<x<4
For maximum volume dV/dx=0
dV/dx=12x2-84x+104=0
3x2-28x+26=0
x=1.046 or x= 8.29
but 0<x<4, so x=1.046
maximum volume=4x3-42x2+104x wherex=10.46
=67.41 ft3
b) length=s
breadth=s
when square of side x is removed,
new length=s-2x
new breadth=s-2x
height=x
volume=(s-2x)2x
for maximum volume dV/dx=0
2(s-2x)(-2)(x)+(s-2x)2=0
-4x+s-2x=0
s=6x
x=s/6
max volume=(s-2x)2x
=(s-s/6)2(s/6)
=(25s2/36)*(s/6)
=25s2/216
Maximum volume is 25s2/216
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