Optimisation of open top tin problem

[JacDevo]

Here is the question:
Consider the following rectangular piece of tinplate. An open top cake tin is to be made by cutting a square (xcm by xcm) from each corner.

+-+--------||-------+-+
| |x               x| |
+-+                 +-+
|x                   x|
|                     |
+-+                 +-+
| |x               x| |
+-+--------||-------+-+

*sorry, I wasn't sure how to insert the picture so I attempted to draw it.

Develop a conjecture about the relationship between x (the cut to be made for the square) and the length of each side of the rectangle such that the cake tin has a maximum volume.

The sides of the rectangle are in a ratio r:s. Consider a rectangle where one side is twice the length of the other (i.e. 2:1). Find the value of x that gives the maximum volume for the cake tin. Repeat this process for rectangular tin plates with sides in at least two other ratios.
Hint: find exact solutions for x (i.e. use the quadratic formula).

dimensions:

|<-----r cm---------->|___
+-+--------||-------+-+ ^
| |x               x| | |
+-+                 +-+ |
|x                   x| s cm
|                     | |
+-+                 +-+ |
| |x               x| | |
+-+--------||-------+-+_v_

My progress so far:
I have been able to find the equation for the volume, which is:
V(x) = x(L-2x)(W-2x)

After substituting the values for L and W (1 and 2 respectively for the 1:2 ratio asked in the question), I found the derivative, which was:

12x^2 - 12x +2

From this, I used the quadratic formula to find the x values, which were:

x = 0.789 and x = 0.211

Since 0.789 is not in the domain and would result in a negative volume,
I concluded that 0.211 was the x value of the maximum volume.

I repeated this with ratios of 1:3, 1:4 and 1:5 and found these maximum x values:

1:3 : x = 0.2257
1:4 : x = 0.2324
1:5 : x = 0.2362

The question asked me to develop a conjuncture about the relationship between x and the length of each side of the rectangle. This section I was unsure about. I attempted to, but I wasn't able to find any relation between the x value and the length of r and s (if there is one). If someone could help with this it would be much appreciated, thanks.

 

[Otis]

Hint: find exact solutions for x

[1:2 : x = 0.211]

1:3 : x = 0.2257
1:4 : x = 0.2324
1:5 : x = 0.2362

Hi. Those solutions are not exact; they are rounded decimal approximations.

Exact solutions will each contain ratios and a radical term; redo, then look for a pattern.

 

[JacDevo]

Thanks for the reply. I placed the x values into a graph and found that even with ratios of 1:1000, the x values approached 2.5 but didn't reach them. I'm not sure what to conclude from this, however.

 

[Otis]

... the x values approached 2.5 but didn't reach them (sic).

I'm not sure what to conclude ...

You were instructed to use the quadratic formula to produce exact solutions. It seems that you didn't.

I pointed this out to you, in my first reply.

If you were to produce the exact solutions and then examine them, I'm thinking that you would detect a pattern.

Did you try to produce exact solutions?

 

[stapel]

Consider the following rectangular piece of tinplate. An open top cake tin is to be made by cutting a square (xcm by xcm) from each corner.

Develop a conjecture about the relationship between x (the cut to be made for the square) and the length of each side of the rectangle such that the cake tin has a maximum volume.

The sides of the rectangle are in a ratio r:s. Consider a rectangle where one side is twice the length of the other (i.e. 2:1). Find the value of x that gives the maximum volume for the cake tin. Repeat this process for rectangular tin plates with sides in at least two other ratios.
Hint: find exact solutions for x (i.e. use the quadratic formula).

dimensions:

|<-----r cm---------->|___
+-+--------||-------+-+ ^
| |x               x| | |
+-+                 +-+ |
|x                   x| s cm
|                     | |
+-+                 +-+ |
| |x               x| | |
+-+--------||-------+-+_v_

How does your exercise differ from this one? Or is it the same one?

My progress so far:
I have been able to find the equation for the volume, which is:
V(x) = x(L-2x)(W-2x)

After substituting the values for L and W (1 and 2 respectively for the 1:2 ratio asked in the question), I found the derivative, which was:

12x^2 - 12x +2

From this, I used the quadratic formula to find the x values, which were:

x = 0.789 and x = 0.211

Since 0.789 is not in the domain and would result in a negative volume,
I concluded that 0.211 was the x value of the maximum volume.

I repeated this with ratios of 1:3, 1:4 and 1:5 and found these maximum x values:

1:3 : x = 0.2257
1:4 : x = 0.2324
1:5 : x = 0.2362

The question asked me to develop a conjuncture about the relationship between x and the length of each side of the rectangle. This section I was unsure about. I attempted to, but I wasn't able to find any relation between the x value and the length of r and s (if there is one). If someone could help with this it would be much appreciated, thanks.

Thank you for showing your results. What were your exact results (that is, the square roots, before you found the decimal approximations)?

Thank you!