Determine the minimum surface area

Question

The base and sides of a container is made of wood panels. The container does not have a lid. The base and sides are rectangular. The width of the container is   . The length is three times the width. The volume of the container is ³ . 

Determine the minimum surface area that this container will have.

Answer 1

So this is a pretty involved solution. First, you have to derive the formula for the surface area. Normally, it's the following:

2(height*width) + 2(width * length) + 2(length * height)

However, since it has no lid, you take away one of the "length * width" surfaces, so you end up with the following:

2(height * width) + 1(width * length) + 2(length * height)

Basically what you do now is replace the dimensions in terms of x and h(the height). What you end up with is the following:

SA = 2xh + 3x^2 + 6xh

Now what you need to do is replace h with something in terms of x. Luckily, we have more information as to what h can represent. We use the volume formula.

V = (length) * (width) * (height)

588 = (3x) * (x) * (h)

Then you solve for h to get h = 588/(3x^2).

Then go back to the surface area formula to replace h with what we just found.

SA = 3x^2 + 8x * (588/(3x^2))

SA = 3x^2 + 1568/x

Now you take the derivative in order to get the width that gives the minimum area, since x is width.

SA' = 6x - 1568/x^2

Set it equal to 0 and solve for x. Get a common denominator and only solve for the top.

6x - 1568/x^2 = 0

6x^3/x^2 - 1568/x^2 = 0

(6x^3 - 1568)/x^2 = 0

6x^3 - 1568 = 0

6x^3 = 1568

x^3 = 261.33

x = 6.39

We're not done yet though. We have to put this x back into the surface area equation to get the minimum surface area.

SA = 3x^2 + 8x * (588/3x^2)

SA = 3(6.39)^2 + 8(6.39) * (588/3(6.39)^2)

SA = 367.88 cm^2

Now we're done.