Question

A box with a square base and open top must have a volume of 237276 cm^3. We wish to find the dimensions of the box that minimize the amount of material used. Find a formula for the surface area of the box in terms of only x, the length of one side of the square base.[Hint: use the volume formula to express the height of the box in terms of x.]

Answer 1

Answer:

$A(x)=\dfrac{x^3+949104}{x}$

Step-by-step explanation:

Given a box with a square base and an open top which must have a volume of 237276 cubic centimetre. We want to find a formula for the surface area of the box in terms of only x, the length of one side of the square base.

Let the side length of the base =x

Let the height of the box =h

Since the box has a square base

Volume,

$V=x^2h=237276\\\\h=\dfrac{237276}{x^2}$

Surface Area of the box = Base Area + Area of 4 sides

$Area, A(x,h)=x^2+4xh$

Substitute h derived above into A(x,h)

$Area=x^2+4x(\frac{237276}{x^2})\\\\A(x)=x^2+\dfrac{949104}{x}\\\\A(x)=\dfrac{x^3+949104}{x}$

Therefore, a formula for the surface area of the box in terms of only x, the length of one side of the square base is:

$A(x)=\dfrac{x^3+949104}{x}$