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.
Attempt 1: a= 20 b= 5
Attempt 2: a= 10 b= 5
Attempt 3: a= 40 b= 20
The Attempt at a Solution
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.
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.
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
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