Volume of an open top box?

Question

An open-topped box with square base (as in the figure at the right) is to be built for $48. The sides of the box will cost $3 per square meter, and the bottom of the box will cost $12 per square meter. If the edge-length of the square base is "x", express the VOLUME V of the box as a function of "x" by entering integers. Hint: use the first two sentences to express the height of the box in terms of "x".

Answer 1

let the height be y

the base area = x^2, costing 12x^2

sides would be 4xy and would cost 12xy

total cost = 12x^2+12xy=48

or x^2+xy=4

or x(x+y)=4

or x+y=4/x

this gives y=4/x - x



Volume V=x^2*y = 4x-x^3

Answer 2

Whether or not the box is opened top or closed shouldn't make any difference in the volume(unless the top has some sort of thickness). However, the surface area WOULD change because you have one extra side to account for.

Answer 3

x = side of the base

h = hieght of the box

C = cost



C = 12 (area of the base) + 3 (perimeter of the base) * height

C = 12x² + 3(4x)h

C = 12x² + 12xh



given the total cost to construct the box is $48

48 = 12x² + 12xh

48 - 12x² = 12xh

h = (48 - 12x²)/(12x)

h = 4/x - x



V = x²h

V = x² (4/x - x)



you can leave like that or you can distribute. Hope it helps!