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 from 1000sq ft of material.
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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 –...
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The surface material = 1000sq ft.
The container has has a square base . So the sides of the base be x and let h be the height.
Then the area of the base = x^2. sq feet.
The are of the four lateral surfaces = 4xh.
So the total surface = x^2+4xh = 1000.
So h = (1000-x^2)/4x
Volume of the container V(x) = x^2h = x^2(1000-x^2)/4x.
V(x) = 250x- (1/4)x^3.
To maximise V(x):
V'(x) = 0 gives V'(x) = 250 - (3/4)x^2.
So x^2= 250*4/3.
x = sqrt(1000/3) =18.26.
Also V''(x) = -(6/4)x . V''(18.26) = - (6/4)(18.26 < 0.
Therefore V(x) is maximum for x =
sqrt(1000/3) = 18.22674 ft and h = 9.1287 ft.
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