Mathematics GCSE coursework
The open box problem
An open box is made from a sheet of card
Identical squares cut off the four corners of the card
The card is then folded along lines to make an open box.
The main aim of the activity is to determine the size of the square cut which makes the volume of the box as large as possible for any given square sheet of card.
Part 1
I have drawn tables for my squares. I have tested four squares; 5cm square, a 10cm square, a 20cm square, a 40cm square. I have decided against trial and improvement. As this method can is time consuming so I have used gone up in 0.5cm each time for the size of x (the cut out size) then calculated the size of v (volume). In one table of values I have gone to 5 decimal places to prove my relationship is correct. The volume also depends on the size of a (the length of the square before the cut out.) which is represented by the letter l.
After I have drawn the tables I will analyze the graphs. I will put the graphs under the table so on the next few pages there will be a table of values showing the different volumes depending on the cut out of the square then a graph giving us a clear picture of the maximum value. You will see this process repeated four times for each of my tested squares.
After doing these stages I will try and work out a relationship between the cut out of the square and the volume. To do this I will generalise the formula then use a method called differentiation. After I have done this I will input the letters and numbers into the quadratic formula and work out the relationship.
Throughout this piece of coursework I will be using this formula
v=x(l-2x) 2
So if the length of the square was 10 and the size of the cut of was 2 then I would sub these figures into the formula as shown below:
The open box problem
An open box is made from a sheet of card
Identical squares cut off the four corners of the card
The card is then folded along lines to make an open box.
The main aim of the activity is to determine the size of the square cut which makes the volume of the box as large as possible for any given square sheet of card.
Part 1
I have drawn tables for my squares. I have tested four squares; 5cm square, a 10cm square, a 20cm square, a 40cm square. I have decided against trial and improvement. As this method can is time consuming so I have used gone up in 0.5cm each time for the size of x (the cut out size) then calculated the size of v (volume). In one table of values I have gone to 5 decimal places to prove my relationship is correct. The volume also depends on the size of a (the length of the square before the cut out.) which is represented by the letter l.
After I have drawn the tables I will analyze the graphs. I will put the graphs under the table so on the next few pages there will be a table of values showing the different volumes depending on the cut out of the square then a graph giving us a clear picture of the maximum value. You will see this process repeated four times for each of my tested squares.
After doing these stages I will try and work out a relationship between the cut out of the square and the volume. To do this I will generalise the formula then use a method called differentiation. After I have done this I will input the letters and numbers into the quadratic formula and work out the relationship.
Throughout this piece of coursework I will be using this formula
v=x(l-2x) 2
So if the length of the square was 10 and the size of the cut of was 2 then I would sub these figures into the formula as shown below: