A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 450 cubic centimeters of soup. The sides and bottom of the container will be made of styrofoam costing 0.03 cents per square centimeter. The top will be made of glued paper, costing 0.08 cents per square centimeter. Find the dimensions for the package that will minimize production cost.

dimensionsof the microwave cup-of-soup package are:Radius= 3.39cm, andHeight= 9.27cm.differentiation.cylinder.volumecan be shown as, V = πr²h, where r is itsradiusin cmheightin cm.volumeis given to be 450 cm³.costof the microwaveable cup-of-soup package is given as:areaof the bottom = πr².costof the bottom = πr²(0.03) cents.areaof the side = 2πrh = 2πr{450/(πr²)}. {Taking h = 450/(πr²) from (i)}.costof the side = 2πr{450/(πr²)}(0.03) cents.areaof the top = πr².costof the top = πr²(0.08) cents.cost functioncan be written as:cost function.Differentiatingboth sides in (ii) with respect to theradius,inflections, we equate this to zero, to get,costC, is minimum or maximum at the point, we calculate thesecond order differential, and if it’s greater than 0, then we get a minimum and vice-versa.differentiatingboth sides of (iii), with respect to the radius r, we get:radiusat which thecostis minimized is 3.393 cm.dimensionsof the microwave cup-of-soup package are:Radius= 3.39cm, andHeight= 9.27cm.differentiation.differentiationat