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Optimisation question? maths question?

Hi..can someone answer this question step by step so i can understand how to do it?...it is decided to investigate the manufacture of a silo capable of of holding 10 000 m^3 of grain. The silo must be in the shape of a cylinder with a hemispherical roof. The grain is stored only in the cylindrical part, not in the roof (hemisphere section). The hemispherical roof costs twice as much per square metre to manufacture as the cylindrical part. What dimensions should the silo have in order to minimise the costs of manufacture?...the dimensions are 'r' (radius of cylinder) and 'h' (height of cylinder)...10 POINTS FOR BEST ANSWER...all answers are appreciated...thanks

Answer:

Let: V - grain volume a - cost to build 1 sq.m of cylindrical part Volume of cylindrical part is: pi*r^2*h = V So h = V / (pi*r^2) Total cost to build the silo is: Cost = pi*r^2*a (bottom of the silo) + 2*pi*r*h*a + 2*pi*r^2*2a (semi-spherical part cost twice) = = a*(5*pi*r^2 + 2*pi*r*h) = = a*(5*pi*r^2 + 2*pi*r*V / (pi*r^2)) = = a*(5*pi*r^2 + 2*V/r) You need to find minimum of the function dCost/dr = a*(10*pi*r - 2*V / r^2) = 0 r = (V/(5*pi)) ^ (1/3) = ... h = V / (pi*r^2) = ...

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