Question

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 0.06 cents per square centimeter.Find the dimensions for the package that will minimize production cost. Helpful information:h: height of cylinder, r: radius of cylinderVolume of a cylinder: V=πr²hArea of the sides: A= 2πrhArea of the top/bottom: A=πr²

Answer 1

Answer:

r = 3.628 cm

h = 10.88 cm

cost = 11.16 cents

Step-by-step explanation:

we know the total volume = 450 cc

then pi * r ^ 2 * h = 450

solved for h:

h = 450 / (pi * r ^ 2)

h = 143.31 / r ^ 2

now the total cost of packing = 2 * 3.14 * r * h * 0.03 + 3.14 * r ^ 2 * 0.03 + 3.14 * r ^ 2 * 0.06

= 0.1884 * r * h + 0.0942 * r ^ 2 + 0.1884 * r ^ 2

= 0.1884 * r * h + 0.2826 * r ^ 2

putting the value of h in the previous equation we obtain

total cost = 27 / r + 0.2826 * r ^ 2

we have to minimize this to differentiate it and equate it with 0 :

-27 / r ^ 2 + 0.5652 * r = 0

r ^ 3 = 27 / 0.5652

r ^ 3 = 47.77

r = 3,628 cm

replacing:

we get h = 450 / (3.14 * 3,628 ^ 2) = 10.88 cm

the total cost can be calculated for the equation

we get the total cost = 27 / 3,628 + 0.2827 * 3,628 ^ 2 = 11.16 cents