So…
New cuboids dimensions = Length = 22m
Width = 10m
Height = 5m
Volume = 10m × 22m × 5m
= 1100m3
S. Area = (2×10×5)+(2×22×5)+(2×10×22)
= 100 + 220 + 440
= 760m3
Mass = 7800kg/m3 × 1100m3
= 8’580’000kg
Volume = 10m × 27m × 5m
= 1350m3
S. Area = (2×10×5)+(2×27×5)+(2×10×27)
= 100 + 270 + 540
= 910m3
Mass = 7800kg/m3 × 1350m3
= 10’530’000kg
This proves that whatever percentage one dimension is increased by, the volume and Mass are increased by the same percentage. This proves that Volume and Mass are directly proportional to the length of any one key dimension. Whether it is the length, width or height that is changed this will be the same.
The task now is to determine whether the Volume, Surface Area and Mass of the cuboid react in the same way if two dimensions are changed.
Changing two key Dimensions by 5%, 10% & 20%
Increasing the Width = 10m
and Height = 5m
5% increase => 10 × 5% = 0.5
100
=> 10m + 0.5m (increase) = 10.5m
So…
New cuboids dimensions = Length = 21m
Width = 10.5m
Height = 5.25m
Volume = 10.5m × 20m × 5.25m
= 1102.5m3
S. Area = (2×10.5×5.25 )+(2×20×5.25)+(2×10.5×20)
= 110.25 + 210 + 420
= 740.25m3
Mass = 7800kg/m3 × 1102.5m3
= 8’599’500kg
This shows that an increase in two dimensions of 10% imposes an increase of 10.25% on the Volume and Mass.
10% increase => 10 × 10% = 1
100
=> 10m + 1m (increase) = 11m
So…
New cuboid dimensions = Length = 20m
Width = 11m
Height = 5.5m
Volume = 11m × 20m × 5.5m
= 1210m3
S. Area = (2×11×5.5 )+(2×20×5.5)+(2×11×20)
= 121 + 210 + 440
= 771m3
Mass = 7800kg/m3 × 1210m3
= 9’438’000kg
35% increase => 10 × 35% = 3.5
100
=> 10m + 3.5m (increase) = 13.5m
So…
New cuboid dimensions = Length = 20m
Width = 13.5m
Height = 6.75m
Volume = 13.5m × 20m × 6.75m
= 1822.5m3
S. Area = (2×13.5×6.75 )+(2×20×6.75)+(2×13.5×20)
= 182 + 270 + 540
= 992m3
Mass = 7800kg/m3 × 1822.5m3
= 14’215’500kg
An increase of 5% for two dimensions produces an increase in Volume and Mass of 10.25%
10.25% is equal to 5% + 5% + (5% of 5%)
= 10% + ( 5 × 5%)
100
= 10% + 0.25% = 10.25%
An increase of 10% for two dimensions produces an increase in Volume and Mass of 21%%
21% is equal to 10% + 10% + (10% of 10%)
= 20% + ( 10 × 10%)
100
= 20% + 1 = 21%
An increase of 35% for two dimensions produces an increase in Volume and Mass of 82.25%
82.25% is equal to 35% + 35% + (35% of 35%)
= 70% + ( 35 × 35%)
100
= 70% + 12.25% = 82.25%
So when two key dimensions are changed by the same amount the percentage change in Volume and Mass can be calculated by adding the two percentages of change together then adding the percentage of the percentage. This could be written as a formula.
Percentage change in volume (v) = 2× Percentage change of dimensions (c) + Percentage of Percentage
v = 2c+( c × c%)
100
Test:
Taking the results from the first sum involving two dimensions, a 5% increase of the width and height of a cuboid measuring, Length = 20m, Width = 10m & Height = 5m.
% Volume increase = 102.5 × 100% = 10.25%
1000
% S. Area increase = 40.25 × 100% = 5.75%
700
% Mass increase = 799’500 × 100% = 10.25%
7’800’000
Using the formula:
A 5% increase of the two dimensions should equal an increase in volume of:
v = 2c+( c × c%)
100
v = 2 × 5% + ( 5% × 5%)
100
= 10 + 0.25
= 10.25% The formula Works…
This investigation has clearly proven that the dimensions of a cuboid are directly proportional to the Volume and Mass of the cuboid.
If the principle of this investigation were applied to an engineering product, that maybe needed to be resized this formula could be very important to the design of the product. For example, it may have an influence on the materials used for production if minimum mass were a design requirement of the product.
Further exploration could discover weather there is a formula that could be used to derive the percentage change in Volume & Mass of a cuboid when the dimensions are changed by different amounts.