Question

A box is to be created using 28 ft² of material. If the box has a square base and top, find the dimensions of the box with greatest volume that may be created. Round dimensions to two decimal places.

Answer 1

Given that box is open-top and square based, So

Length = width = L

height = h

Volume of box is given by

V = L*L*h

Surface area of box will be:

S = 2*L*L + 4*L*h = 2*L^2 + 4Lh = 28 ft^2

h = (14 - L^2)/(2*L)

Using above value

V = L^2*(14 - L^2)/(2*L)

V = 7*L - L^3/2

Now Volume will be minimum when

dV/dL = 0

dV/dL = 7 - (3/2)*L^2 = 0

7*2/3 = L^2

L^2 = 14/3

L = (14/3)^(1/2) = 2.16 ft

h = (14 - L^2)/(2*L) = (14 - 2.16^2)/(2*2.16)

h = 2.16 ft

So, length & width of box = 2.16 ft and height of box = 2.16 ft

Now, greatest volume will be:

V = L^2*h = 2.16*2.16*2.16

V = 10.08 ft^3

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