Question:

a) We have a folded box shaped like a rectangle with the length b and width a. The box has four squares that we fold up, these squares are the height of the box x. Choose a couple of values for length b and width a and decide what x has to be in order for the box to have the largest volume.

b) Produce a formula that enables you to calculate what x has to be in order to get the largest volume when only given the length a and width b.

My attempt to a solution:

a)

1. First i got to make a function that describes the volume of the box
2. Derive the function in order to find the maximum point of the function
3. Find out which x-values that can't be a part of the solution (this is where my first problem lies) for example that x > 0 but can't be larger than...
4. If both x values are a part of the solution, then find out which one lies at a maximum point and finding the y coordinate by inserting the x value into the normal function.

Length: b-2x
Width: a-2x
Height: x

Attempt 1:

y= x(b-2x*a-2x)
y= x(ab + b*-2x + a*-2x +4x^2)

I then chose a = 20 b = 5 (measured in cm) this gave me

y= x(20*5 +5*-2x -20*-2x +4x^2)
y= 4x^3 -50x^2 +100x
y´= 12x^2 -100x +100
y´= x^2 - 25/3x + 25/3

When solving the equation using the quadratic formula i get x= 1,16 and x= 7,16 and when searching for the maximum point i get that when x= 1,16 we're at a maximum point which gives us y = 54,94. The maximum volume being 54,94cm^3 when x = 1,16.

Attempt 2:

a= 10
b=5

On attempt 2 i got x = 3,94 and x= 1,05 and the maximum point was at x= 1,05 and y= 54,5

Attempt 3:

a= 40
b= 20

On attempt 3 i got x= 15,77 x= 4,22 and the maximum point was at x= 4,22 and y= 1539,59.

b) This is where my problem lies, creating a formula where i only need a and b to get what x needs to be in order to get the maximum volume

Thanks in advance Mathmen

is awfully ambiguous. If you did not know where you came from you would not be able to interpret the expression correctly. The right way should be

y= x(b-2x)*(a-2x)

What you wrote has a completely different meaning. The next one

y= x(ab + b*-2x + a*-2x +4x^2)

I would write either as

y= x(ab + b*(-2x) + a*(-2x) +4x^2)

or as

y= x(ab - 2bx - 2ax +4x^2).

So this is your function (get it by multiplying out):

y = 4x³ - 2(a + b)x² + abx.

To find the extremes you differentiate and equate to 0:

y' = 12x² - 4(a + b)x + ab = 0.

This is a quadratic equation with roots

x = {4(a + b) ± sqrt(16(a + b)² - 48ab)} / 24

Simplify:

x = {(a + b) ± sqrt(a˛ - ab + b˛)} / 6

For you to determine which of these is a maximum.