How do i solve this? Should i use limits?

Question

A rectangular box without a lid (as shown below diagram) is to be made from 27 cm²

cardboard. Find the maximum volume of such a box.

Answer 1

It's hard too know how to answer this problem because I'm not sure where you are in the "Calculus curriculum". If this is the only information given, then the problem is a multi-variable problem requiring Calc 3 type methods. However, we can do this problem using Calc 1 methods if we make a simple assumption. Let's initially assume that a = b.

The equation for the surface is:

SA = ab + 2ac + 2bc and, letting a = b it becomes:

SA = a² + 2ac + 2ac = a² + 4ac and since we know SA = 27, we can say:

27 = a² + 4ac

27 - a² = 4ac

c = (27 - a²)/4a

The volume is V = abc but since a = b and c = (27 - a²)/4a, we can say:

V(a) = a²(27 - a²)/4a

V(a) = a(27 - a²)/2

V(a) = 27a/2 - a³/2

To maximize, take the derivative and set it equal to zero:

V'(a) = 27/2 - 3a²/2

27/2 - 3a²/2 = 0

27 - 3a² = 0

9 - a² = 0

a² = 9

a = 3

Since we assumed a = b, then b = 3

c = (27 - a²)/4a = (27 - 3²)/4•3 = 18/12 = 1.5

So the volume = 3•3•1.5 = 13.5 cm³

Now we can test our assumption by letting a and b NOT be equal.

SA = ab + 2ac + 2bc

27 = ab + 2ac + 2bc

27 - 2bc = ab + 2ac

27 - 2bc = a(b + 2c)

a = (27 - 2bc)/(b + 2c)

So Volume is:

V = abc

V = [(27 - 2bc)/(b + 2c)]bc

Let's let b = 2.9 and then let b = 3.1 but maintain c = 1.5 and see what happens to the volume:

For b = 2.9:

V = [(27 - 2•2.9•1.5)/(2.9 + 2•1.5)]•2.9•1.5 = 13.4924

For b = 3.1

V = [(27 - 2•3.1•1.5)/(3.1 + 2•1.5)]•3.1•1.5 = 13.4926

So, we indeed see that the volume of 13.5 cm³ is a maximum.

You can repeat the same thing by testing different values of c (rather than maintaining it at 1.5) and you get similar results.

Using multi-variable calculus, you would take partial derivatives but it gets a lot more complicated. If that's what you are working on, my apologies.