A genetically modified potato variety has been developed with an average potato

Question

A genetically modified potato variety has been developed with an average potato size of 105 mm x 72 mm x 65 mm (standard deviation of 6.7 mm x 3.4 mm x 2.5 mm respectively). HuskerFoods has identified a distributor of this new potato variety that will charge 25% more than the standard potato variety distributor. Use the relationships developed between potato dimensions and surface area to estimate the surface area of the new potato variety. Provide a recommendation to HuskerFoods on whether or not they should explore the new potato variety. Explain your reasoning.

Explanation / Answer

given dimensions of potato
2a = 105 mm = 0.105 m
2b = 72 mm = 0.072 m
2c = 65 mm = 0.065 m

assuming the potato to be an ellipsoid
consider a potato double ther given dimensions
Surface area, S = 4*pi((ab)^1.6 + (bc)^1.6 + (ca)^1/6)^(1/1.6)/3^(1/1.6)
S = 8.0756*10^-2 m^2

Volume, V = 4*pi*abc/3 = 2.05837*10^-3 m^3

now lets use the original dimensions
a = 0.0525 m
b = 0.036
c = 0.0325
surface area s = 2.0189*10^-2 m^2
volume v = 2.5729*10^-4 m^3

consider, (S - s)*100/s = 300
consider (V - v)*100/v = 700

so a 100 percent increase in dimension causes surface area to increase by 300 pc and volume to increase by 700 pc
hence the new potato variety which is bigger than the normal version pack more volume htan surface area
as potatoes get infected from ther surface, reducing surface area while increasing voloume increases the weight of the potato to be szold per unit srface area infected ( assuming as constant percent of surface area that gets effected)
hence such variety of potato is more profitabel for sale and should attract lesser selling charges than the norml variety

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