A rectangular sheet of metal measures 50 cm by 40 cm. Equal squares of side x cm are cut from each corner.?

Question

The sheet is then folded up to make a tray of depth x cm. What is the domain possible values of x? Find the value of x which maximises the capacity of the tray.

Who can help me with this maths problem?

Answer 1

Your first step is to make a diagram. Since x by x corners are cut off, you'll have to represent this by drawing a rectangle, and labeling the x by x squares that you're going to remove appropriately.

Since we cut the corners off, you'll notice that the length will be equal to (50 - 2x) and the width will equal (40 - 2x). Again, it is much easier to use a diagram and you'll be able to tell right away how to get the length and width.

Therefore, since we want to maximize the capacity of the tray (i.e. the volume), and volume is equal to length times width times height (the height is just x), we get

V = (50 - 2x) (40 - 2x) (x)

Which we'll just make a function of, i.e
V(x) = (50 - 2x) (40 - 2x) (x)

Eventually we want to find the derivative and make it 0, so we have to expand this.

V(x) = [200 - 180x + 4x^2](x)
V(x) = 200x - 180x^2 + 4x^3

Now, we take the derivative.
V'(x) = 200 - 360x + 12x^2
and make it 0.
0 = 200 - 360x + 12x^2

To make things simpler for factor, let's order this in descending x power.
12x^2 - 360x + 200 = 0
Let's also divide by the greater common factor, 4.

3x^2 - 90x + 50 = 0
Using the quadratic formula,

x = (90 +/- sqrt(8100 - 600))/6
x = (90 +/- sqrt(7500))/6
x = (90 +/- 10 sqrt(75))/6
x = (90 +/- 50 sqrt(3))/6
x = (45 +/- 25 sqrt(3))/3

Therefore, x = 9 + 25/3 sqrt(3) or x = 9 - 25/3 sqrt(3)

We discard the second value because it's negative, so our only answer for x is

x = 9 + 25/3 sqrt(3)

Answer 2

Since we're cutting out squares of size x from each corner, we need two xs to be able to fit in either direction, so x is between 0 and 20 cm.

The volume then becomes (50-2x)(40-2x)x.
Expanding that gives 4x^3 - 180x^2 + 2000x.
Differentiating: 12x^2 - 360x + 2000 = 0
Or 3x^2 - 90x + 500 = 0.
Use the quadratic formula to find x = 22.637 or x = 7.362.
Since x is less than 20 cm, x must be 7.362 cm, to 3dp.