The cut out square which gave the biggest volume was 3cm by 3cm.
Summary Table
0.16 is 1/6.
The size of the cut out square is approximately 1/6 of the whole square.
From looking at my results table, I predict that the biggest volume for a 14 x 14 square will be completed by cutting a square which is 1/6 of 14.
14cm x 14cm square
1/6 of 14=2.33
I will now check if my prediction is correct by verifying if it is the biggest volume.
Looking at the results in my prediction table, it proves that my prediction was correct; the cut out square with the largest volume was 2.33 x 2.33.
I will now investigate rectangular pieces of card; I will begin by examining rectangles where the length is twice the size of the width, so the ratio of the width to the length is 1:2.
I have illustrated a diagram below to demonstrate the layout of the rectangle and what I am going to be doing. The parts labelled x will be the cut out squares and will be equal size/height to the other cut out squares.
x x
x x
x x
x x
I will now illustrate a diagram to show what the rectangular box will look like when it is assembled. The parts labelled: X=height, W=width and L=length.
X
W
L
I will look at rectangles of sizes: 14cm x 28cm, 15 cm x 30cm and 16cm x 32cm.
To begin with, I will look at the rectangles with dimensions of 14cm x 28cm.
The square cut out which gave the biggest volume using the formula L x W x H was 2.95 x 2.95.
Secondly, I will look at the rectangle 15cm x 30cm.
The cut out square which gave the largest volume was 3.15 x 3.15 by using the formula L x W H.
Last of all, I am going to look at 16cm x 32cm.
From using the formula L x W x H to calculate, the square cut out which gave the highest volume was 3.4 x 3.4.
Results Table - Maximum volume- ratio 1:2.
Looking at my results, I can tell that the best square to cut the corners from is almost 105/1000 of the rectangle’s length.
30cm x 60cm
I predict that the cut out square will be 6.3 x 6.3.
As I can see from my results table, the cut out square is 6.3 x 6.3; therefore, my prediction was correct.
The cut out square which gave the highest volume was 2.25 x 2.25.
I have been looking at rectangles in the ratio 1:2, I am now going to be investigating rectangles where the width and the length are in the ratio 1:3 so the length is 3 times the size of the width.
I will look at rectangles of sizes: 10 x 30, 12 x 36, and 14 x 42.
I will begin by looking at rectangle of size 10 x 30.
Secondly, I will look at 12 x 36.
The cut out square which gave the largest volume is 2.7 x 2.7.
Lastly, I will look at the rectangle 14 x 42.
The square cut out which gave the biggest volume is 3.15 x 3.15.
Summary Table
From the results in the table, I can see that the best square to cut the corners from is approximately 75/1000 of the whole of the rectangle’s length.
30cm x 90cm
I will predict that the cut out square will be 6.75 x 6.75.
From my prediction table, I can see that my prediction was 100% correct; the cut out square which gave the highest volume is 6.75 x 6.75.
These were my results for the square and my first two rectangles using the ratios: 1:1 and 1:2, 1:3.
This took me a while to calculate the proportion of the square or rectangle which gave the largest volume. I will now be using differentiation to look for the proportion of rectangle which gives the maximum volume.
My teacher suggested researching this method to extend my investigation further, so I will now look at rectangles where the width and length are in the ratio 1:4 and 1:5.
I will look at a rectangle with the ratio 1:4 where the length is 4 times the size of the width and the rectangle I am going to use is 10cm x 40cm.
Length=40-2x
Width=10-2x
Height=x
So the volume is (40-2x)(10-2x)x
Y=(4x²-100x+400)x
Y=4x³-100x²+400x
X 40 -2x
10 400 -20x
-2x -80x 4x²
Equation of the curve
= 12x²-200x+400
a=12
b=200
c=400
x=-b± b² - 4ac
2a
x=-(-200)± ((-200)²-4(12)(400))
2(12)
x=200± 20800
24
So x will equal:
x=200+ 20800 or 200- 20800
24 24
x=(200+144.222051) (200-144.222051)
24 24
=14.34258346 =2.324081208
The size of the corner which gives the largest volume is 2.324081208 or 2.3 (to 2 significant figures); it cannot be possible to cut a corner of size 14.3cm from a rectangle of size 10 x 40.
I am now going to look at rectangles with the ratio 1:5 where the length is 5 times the size of the width. The rectangle I am going to use is 20cm x 100cm.
Length=100-2x
Width=20-2x
Height=x
So the volume is=(100-2x)(20-2x)x
Y=(4x²-240x+2000)x
Y=4x³-240x²+2000x
X 20 -2x
100 2000 -200x
-2x -40x 4x²
Equation of the curve
= 12x²-480x+2000
a=12
b=-480
c=2000
x=-b± √ b²-4ac
2a
x= 480± √(480)²-4(12)(2000)
2(12)
x= 480± √ 134400
24
So x will equal:
x = 480+√134400 or = 480- √134400
24 24
x = (480+366.6060556) (480-366.6060556)
24 24
=32.27525232 =4.724747683
The size of the corner which gives the biggest volume is 4.724747683 or 4.7 (to 2 significant figures); it is unlikely to cut a corner of size 32.3cm from a rectangle which is 20cm x 100cm.
Differentiation for the ratio 1:10
The rectangle I will look at is 8cm x 80cm.
Length= 80-2x
Width= 8-2x
Height = x
Therefore the volume is V = (80 – 2x)(8-2x)x
V = (80-2x)(8x-2x²)
V = 640x – 160x² - 16x² - 4x³
V = 4x³ - 176x² + 640x
Equation of the curve:
dx
dv = 12x² - 352x + 640
a = 12
b = -352
c = 640
x = -b±√b²-4ac
2a
x = 352±√(-352)²-4(12)(640)
2(12)
x = 27.38585602 or = 1.947477314
Differentiation for 1:20
The rectangle I will look at is 5:100
Length = 100-2x
Width = 5-2x
Height = x
So the volume is V = (100-2x)(5-2x)x
V = (100-2x)(5x-2x²)
V = 500x – 200x² - 10x² + 4x³
V = 4x³ - 210x² + 500x
Equation of the curve:
dx
dv = 12x³ - 420x + 500
a= 12
b= -420
c= 500
x = -b±√b²-4ac
2a
x = 420±√-420²-4(12)(500)
2(12)
x = 33.76601775 or 1.233982253
Conclusion
I am now going to collect my results. I will look at the proportion of both length and width, which gives the maximum volume.
Length:
Width:
