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1 vote
A manufacturer wants to make open tin boxes from pieces of tin with dimensions 8 in. by 15 in. by cutting equal squares from the four corners and turning up the sides. Find the side of the square cutout that gives the box the largest possible volume.

0 in.
6 in.
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User Dewaffled
by
8.2k points

1 Answer

5 votes

Answer:


(5)/(3) in

Explanation:

Let x be the side of square.

Length of box=8-2x

Width of box=15-2x

Height of box=x

Volume of box=
L* B* H

Substitute the values then we get

Volume of box=V(x)=
(8-2x)(15-2x)x=(15x-2x^2)(8-2x)


V(x)=120x-30x^2-16x^2+4x^3


V(x)=4x^3-46x^2+120x

Differentiate w.r.t x


V'(x)=12x^2-92x+120


V'(x)=0


12x^2-92x+120=0


3x^2-23x+30=0


3x^2-18x-5x+30=0


3x(x-6)-5(x-6)=0


(x-6)(3x-5)=0


x-6=0\implies x=6


3x-5=0\implies x=(5)/(3)

Again differentiate w.r.t x


V''(x)=24x-92

Substitute x=6


V''(6)=24(6)-92=52>0

Substitute x=5/3


V''(5/3)=24(5/3)-92=-52<0

Hence, the volume is maximum at x=
(5)/(3)

Therefore, the side of the square ,
x=(5)/(3) in cutout that gives the box the largest possible volume.

User Mikyra
by
8.4k points
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