Question
Asked Oct 18, 2019

If an open box has a square base and a volume of 115 in.3 and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction. (Round your answers to two decimal places.)

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Expert Answer

Step 1
Let x be the side of the square base, and h be the height of the box.
Then, the volume of the open box is V length x breadth x height x2h
It is given that the volume of the open box is V 115 in.2. Hence, the height of the box
is obtained as follows
x2h 115
115
2
The surface area of the open box is obtained as follows
2length x breadth +2breadth x height +2length x height
(since the box is open
S
=x2 2xh2xh
= x2 4xh
help_outline

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Let x be the side of the square base, and h be the height of the box. Then, the volume of the open box is V length x breadth x height x2h It is given that the volume of the open box is V 115 in.2. Hence, the height of the box is obtained as follows x2h 115 115 2 The surface area of the open box is obtained as follows 2length x breadth +2breadth x height +2length x height (since the box is open S =x2 2xh2xh = x2 4xh

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Step 2
115
Substitute the value of height h=
in the surface area.
2
x
115
S =x24x
115
= x +4
460
=x
x
In order to optimize the material use, we have to minimize the surface area by finding the
460
with respect to x
derivative of the surface area S =x
d
S'
460
460
= 2x
help_outline

Image Transcriptionclose

115 Substitute the value of height h= in the surface area. 2 x 115 S =x24x 115 = x +4 460 =x x In order to optimize the material use, we have to minimize the surface area by finding the 460 with respect to x derivative of the surface area S =x d S' 460 460 = 2x

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Step 3
0
Calculate the critical point by plugging S
460
2x
0
460
2x
x230
x (230) 6.13 in
115
3.06 in
Hence, the value of height of the open box is obtained as h
(230)
460
with respect to x.
x
Find the second derivative of the function S x
P
S"=
(s)
dx
2x-
dx
920
-2+
help_outline

Image Transcriptionclose

0 Calculate the critical point by plugging S 460 2x 0 460 2x x230 x (230) 6.13 in 115 3.06 in Hence, the value of height of the open box is obtained as h (230) 460 with respect to x. x Find the second derivative of the function S x P S"= (s) dx 2x- dx 920 -2+

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