A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 500 cubic centimeters of soup. The sides and bottom of the container will be made of styrofoam costing 0.04 cents per square centimeter. The top will be made of glued paper, costing 0.07 cents per square centimeter. Find the dimensions for the package that will minimize production cost.

minimum cost is 15.51 cent

Explanation:

given data

volume = 500 cubic centimeter

side cost = 0.04 cents per square centimeter

top costing = 0.07 cents per square centimeter

to find out

minimize production cost

solution

first we find the volume that is

volume = π*r²*h

so here put value and find h in r form

500 = π*r²*h

h = 500 / π*r² ... a

so cost of packing will be

cost = area of side + volume side + volume top

cost = 2π*r*h*0.04 + π*r²*0.04 + π*r²*0.07

here h from equation a

cost = 2π*r * (500 / π*r²) * 0.04 + π*r²*0.04 + π*r²*0.07

cost = 40 / r + 0.346*r² ... b

for minimum put it equal zero and differentiate it

-40 / r² + 0.691 * r = 0

r = 3.86

so from equation a

h = 500 / π*r²

h = 500 / π*3.86²

h = 10.63 cm

so total cost will be by equation b

cost = 40 / 3.86 + 0.346*3.86²

cost = 15.51 cent