Amicrowaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 350 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.

information:

to minimize the cost of the package:

radius:

height:

minimum

x = r = 3,12 cm

h = 11,45 cm

C(min) = 10,09 $

Step-by-step explanation:

Let x be radius of base  and  h height of cylinder

Let A₁ be   Cylinder area of base  +  lateral area

A₁  =  π*x²  +  2*π*x*h        Now V = 350 cm³     350  = π*x²*h

h = 350/ π*x²

Then

A₁  =  π*x²  +  700/x

A₂  area of the top

A₂  = π*x²

Now we write the expression of cost as fuction of x

C(x)  = 0,03 * ( π*x²  +  700/x)   +  0,08*π*x²

C(x)  = 0,03* π*x²  +   21/x   +  0,08*π*x²

C(x)  = 0,11* π*x² +   21/x  

Taking derivatives on both sides of the equation

C´(x)  =  2*0,11*π*x  - 21 / x²

C´(x)  = 0         0,22*π*x  - 21 / x²   =  0

0,22*π*x³  -  21  =  0         0.6908* x³  -  21   = 0

X³ =  21/0,6908               X³ =  30,40

x  =  3,12 cm

And

h  =  350/π*x²        h  =  350 / 30,57        

h = 11,45 cm

C(min)  =  0,11* π*(3,12)² +   21/3,12  

C(min)  =  3,36  + 6,73

example 2 – simplify: step 1: completely factor both the numerators and denominators of all fractions. step 2: change the division sign to a multiplication sign and flip (or reciprocate) the fraction after the division sign; essential you need to multiply by the reciprocal. step 3 : cancel or reduce the fractions.