Question

Calculus

A cylinder shaped can needs to be constructed to hold 350 cubic centimeters of soup. The material for the sides of the can costs 0.02 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.05 cents per square centimeter. Find the dimensions for the can that will minimize production cost.


Volume of a cylinder: V = pi r^2 h

Area of the sides: A = 2pi r h

Area of the top/bottom: A = pi r^2

To minimize the cost of the can:
Radius of the can:
Height of the can:
Minimum Cost

Answer 1

Lets solve for h first.
\[V= \pi r^2h \implies 350 cm^3 = \pi r^2 h \implies h = \frac{ 350cm^3 }{ \pi r^2 } \]

Answer 2

Following

Answer 3

Now we can also find the surface area

Answer 4

by doing the cost function?

Answer 5

\[S = 2 \pi r h+2 \pi r^2\] so solve for this

Answer 6

I did not know this part

Answer 7

You'll have to plug in for h in the second equation and then you can find cost

Answer 8

S=2πr(350cm3πr2)+2πr

Answer 9

Idk how to do the Pi smybol i tried copy and paste

Answer 10

Yeah I can't see it, only shows question marks.

Answer 11

so sub the H with the height equation?

Answer 12

Yes or you can just find the integer h but, \[S = 2 \pi r (\frac{ 350 cm^3 }{ \pi r ^2 })+2 \pi r^2\]

Answer 13

ok which is what I got.

Answer 14

Yeah just simplify it and what do you get?

Answer 15

Then we can find the cost of the materials

Answer 16

2(pi)r*350

Answer 17

Not quite

Answer 18

\[S = 2\frac{ 350 cm^3 }{ r } + 2 \pi r^2\]

Answer 19

I subtracted the 2(pi)r and then multiplied by (pi)r^2
to get 2(pi)r^3

Answer 20

Now the cost\[C = 0.02\left( \frac{ 700 }{ r } \right) + 0.05\left( 2 \pi r^2 \right)\]

Solve for this and find the derivative

Answer 21

Once you find the derivative set it to 0 and solve for r. So you'll have h and r then go back and plug all the numbers in for h and you'll have the minimum cost.

Answer 22

Oh ok, that makes sense now, so what did I do wrong with the equation to get to where you got?

Answer 23

I think you just plugged in the wrong numbers, it's hard to tell as it's not in LaTeX, so I really didn't read your equation haha. If you drew it out, I can see what you did wrong :d

Answer 24

Oh I see, you were multiplying when you should be dividing.

Answer 25

Let me give it a go

Answer 26

Take your time

Answer 27

\[2\pi(r)(h)+2\pi(r)=2\pi(r)(350(\div)\pi(r)^2)-2\pi(r)^2\]

Answer 28

So here I subtracted the 2pi(r) to bring over to the left, then multiplied by pi(r)^2 to get 2pi(r)^3 on the left and 2pi(r)2 on the left

Answer 29

Why are you doing that?

Answer 30

Those are two separate equations you don't have to do that, I was just showing solve for one equation (finding h) then plug that into the surface area equation.

Answer 31

You over complicated the problem :), no worries though, do you understand now?

Answer 32

Oh Ok yeah I understand now, the benefits of calculus is me always doing that haha but yes I now understand thank you very much!

Answer 33

Np :)