## burhan101 Group Title 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. one year ago one year ago

1. zepdrix Group Title

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 Group Title

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

3. zepdrix Group Title

$\large 500=\pi r^2h$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 Group Title

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 Group Title

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

6. burhan101 Group Title

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

7. burhan101 Group Title

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

8. zepdrix Group Title

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 Group Title

yes that's what I took the derivative of

10. zepdrix Group Title

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 Group Title

Nope, i made a mistake in the quotient rule

12. burhan101 Group Title

How do i solve for 'x' now?

13. zepdrix Group Title

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

14. burhan101 Group Title

oh yes okay !

15. burhan101 Group Title

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

16. dan815 Group Title

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

17. zepdrix Group Title

$\huge 0=\frac{ 4(\pi r^3-250) }{ r^2 }$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 Group Title

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

19. zepdrix Group Title

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

20. dan815 Group Title

oh i see

21. dan815 Group Title

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

22. zepdrix Group Title

heh