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Area & Volume Exploration – Component proportional changes

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Introduction

Area & Volume Exploration - Component proportional changes Question 1: How do the Volume, Surface Area and Mass of your component vary when two key dimensions are changed but the length remains the same? Changing one key Dimension by 5%, 10% & 20% Increasing the length: 20m 5% increase => 20 ? 5% = 1 100 => 20m + 1m (increase) = 21m So... New cuboid dimensions = Length = 21m Width = 10m Height = 5m Volume = 10m ? 21m ? 5m = 1050m3 S. Area = (2?10?5)+(2?21?5)+(2?10?21) = 100 + 210 + 420 = 730m3 Mass = 7800kg/m3 ? 1050m3 = 8'190'000kg This clearly shows that when the length is increased by 5% the Volume and Mass are also increased by 5%. This indicates that the Volume and Mass are directly proportional to the length. The Surface Area would not appear to be directly proportional to the length as it does not increase by 5%. Further exploration is needed to confirm that this proportional increase is not a one off event. I predict that the same will happen and the percentage increase will be the same for Length, Volume and Mass. ...read more.

Middle

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) ...read more.

Conclusion

% 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. ...read more.

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