Express the total cost of the cylinder as a function of r?

Question

The volume of the cylinder is 3900 cubic feet.

The bottom of the cylinder is reinforced steel, costing $35 per square foot, whereas the sides and top cost just $10 per square foot. Express the total cost of the cylinder as a function of r. 

Answer 1

Hello Alexes,

 

This is basically a plan to built a cylinder to storage of something that you do not want to spill, so make the bottom heavy at an increased cost. Let C(r) be the cost of the cylinder

 

C(r) = (bottom area)*(cost of bottom area) + (top area + sides (lateral) area)*(cost of sides and top)

 

cost of bottom area    = $35

cost of sides and top   = $10

                        

The constant Pi =π

 

A                    C=2π r                B

----------------------------------------

|                                                |

|                                                |

|                                                |

|                                                | h
|                                                |

|                                                |

----------------------------------------

C                                                D

Looking at the above drawing, it is a rectangular sheet you to make the cylinder by welding points A and B,(try with a sheet of paper). AC will form the bottom and BD will be the top.

AC = BD = height of the cylinder = h

AB = CD = circumference of the circle that forms the bottom and the top

 

Area of top = πr²

Area of bottom = πr²

sides area = C*h = 2πrh

                       

C(r) = (πr²)*35 + (πr² + 2πrh)*10

    

       = 35πr²  + 10πr² + 20πrh

 

We need to find the height h, volume of the cylinder is given as 3900 cubic feet

 

V = πr²h

3900 = πr²h which gives  h =3900/(πr²) pllug this in the equation of the cost

 

C(r) = 45πr² + 78000/r