# 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.

justaguide | Certified Educator

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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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neela | Student

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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