Question

A chemical refinery needs a vat. They want the vat to be a rectangular prism (with square bases) that has a maximum volume of 1,000 cubic feet. They want to have it constructed to use the least amount of material. What should the lengths of the bases and sides be? (Note: One of the bases is the lid.)

Answer 1

For least material to be used lengths of square base and sides = 10 units.

Explanation:

Let the lengths of the square base and the sides = x feet, x feet and y feet

Area of the square base = x² feet

Volume of the rectangular prism = Area of the square base × Height

= x²y cubic feet

1000 = x²y

y =
`(1000)/(x^2)` -------(1)

Material used in the prism = Surface area of the rectangular prism

= 2(lb + bh + hl)

Here, h = height of the prism

l = length of the base

w = Width of the base

Material to be used (S) = 2(xy + x² + xy) - Area of lid

S = 2(x² + 2xy) - x²

S = x² + 2xy

Now by substituting the value of y from equation (1),

S = x² +
`2x((1000)/(x^(2) ))`

= x² +
`(2000)/(x)`

For least amount of material used,

We will find the derivative of the given function and equate it to zero.

S' = 2x -
`(2000)/(x^(2) )`

2x -
`(2000)/(x^(2) )` = 0

2x³ = 2000

x³ = 1000

x = 10 feet

From equation (1),

y =
`(1000)/((10)^2)`

y = 10 feet

Therefore, for least amount of the material used lengths of square base and sides will be 10 feet.