Monday, October 29, 2012

Find the dimensions of the container with the greatest volume.The container, with a square base, vertical sides and an open top, is to be made...

A box with a square base is made using 1000 ft^2 of material.
Now we have to find the maximum possible volume of the box.


We know
that the volume of the box would be V=x^2*y, where x is the side of the square bottom and y is
the height.


The surface area of material used is x^2 + 4xy which is
equal to 1000 cm^2.


x^2 + 4xy =
1000


=> y = (1000 –
x^2)/4x


Substituting this in the expression for volume we get V =
x^2*(1000 – x^2)/4x


Now we have to maximize
V


V = x^2*(1000 – x^2)/4x


=> V =
x*(1000 – x^2) / 4


V’ = (1/4) [x* (-2x) + 1000 -
x^2]


=> (1/4) [- 2x^2 + 1000 –
x^2]


=> (1/4) [1000 –
3x^2]


Equate V’ to 0


=> (1/4)
[1000 – 3x^2] = 0


=> 3x^2 =
1000


=> x^2 = 1000 / 3


=>
x = sqrt 1000/3 [we don’t need the negative root]


=> x= 18.25
ft


We see that V’’ = -3x/2 which is negative, therefore V is maximum
for this value of x.


Now y = (1000 –
x^2)/4x


=> y = (1000 – 1000/3) / (4* sqrt(1000 /
3)


=> 2000 / 3*4 sqrt
1000/3


=> 9.128
ft


The required maximum volume is achieved with the
base having sides of 18.35 ft and the height 9.128 ft.

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