Question

A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 350 cubic

Answer 1

Answer:

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