## burhan101 2 years ago A cylindrical can is made to hold 500 mL of soup. Determine the dimensions of the can that will minimize the amount of metal required.

1. zepdrix

Volume of a Cylinder:$\large V=\pi r^2h$ They want us to minimize the amount of metal. The amount of metal is the Surface Area. Surface Area of a Cylinder (If I'm remembering this correctly):$\large A=2\pi r^2+2 \pi r h$

2. zepdrix

They want us to minimize the surface area, given a constraint on the volume.

3. zepdrix

$\large 500=\pi r^2h$Solving for h gives us,$\large \color{orangered}{h=\frac{500}{\pi r^2}}$ We'll plug this into our Area formula. $\large A=2\pi r^2+2 \pi r \color{orangered}{h} \qquad\rightarrow\qquad A=2\pi r^2+2 \pi r \color{orangered}{\frac{500}{\pi r^2}}$

4. zepdrix

Simplify it down and then take the derivative of your area function. Then setting it equal to zero will allow you to find critical points, namely the value of r that will minimize the area.

5. burhan101

$\huge A=2 \pi r^2+1000$

6. burhan101

$\huge 0=4\pi r-1000r$

7. burhan101

am i doing it right so far (that is the derivative)

8. zepdrix

I dunno if you simplified that correct for area. Shouldn't you get something like this? Remember the bottom r is squared. $\huge A=2 \pi r^2+\frac{1000}{r}$

9. burhan101

yes that's what I took the derivative of

10. zepdrix

Hmm, your second term looks a little off. We should get something like this, $\huge A'=4\pi r-\frac{1000}{r^2}$ Need to see steps?

11. burhan101

Nope, i made a mistake in the quotient rule

12. burhan101

How do i solve for 'x' now?

13. zepdrix

For r? Set equal to zero as you did. Then get a common denominator, turn it into one big fraction.

14. burhan101

oh yes okay !

15. burhan101

$\huge 0=\frac{ 4(\pi r^3-250) }{ \pi^2 }$ how do i solve for r now @zepdrix

16. dan815

wut u mean its simple, solve for it, remember pi is just some constant

17. zepdrix

$\huge 0=\frac{ 4(\pi r^3-250) }{ r^2 }$Multiply both sides by r^2 giving us,$\huge 0=4(\pi r^3-250)$Then divide both sides by 4, and solve! :)

18. dan815

^ he means multiply by pi^2 but u get the point

19. zepdrix

no, he put pi^2 on the bottom as a mistake.

20. dan815

oh i see

21. dan815

i was wondering why hed ask for help to solve that xD

22. zepdrix

heh