Question

A cylindrical can is to hold 1/2 cup of tomato sauce. What are the dimensions of its diameter and height in centimeters, to minimize the amount of metal to make the can?

V=(pi*d^2*h)/4
A=(pi*d^2)+pi*d*h
1qt=946cc

Answer 1

this is a tricky one....

Answer 2

are thos answers? or formulas to apply?

Answer 3

formulas it's a related rates question

Answer 4

yes, optimizations and such

Answer 5

yes they want D, H, and Amin

Answer 6

V = 1/2 cup, soo;

.5 = (pi) (r^2) (h) right?

Answer 7

Yes

Answer 8

A = 2 circles and a rectangle :)

Answer 9

haha :)

Answer 10

A = 2pi (r^2) + h(2pi r)

Answer 11

use the volume to calibrate either h or r

h = .5/pi r^2 seems easiest right?

Answer 12

substitute that "value" into the Area formula and then derive

Answer 13

which value the 946cc?

Answer 14

the .5 cups...worry about conversions afterwards

Answer 15

oh okay gotcha

Answer 16

so...\[h = \frac{1}{2\pi r^2}\] right?

Answer 17

A = 2pi (r^2) + h(2pi r) sooo

\[A = 2\pi r^2 + \frac{2\pi r}{2\pi r^2}\]

Answer 18

I did get that for A

Answer 19

\[A = 2\pi r^2 + {1 \over r}\] correct?

Answer 20

yes

Answer 21

now derive :) you know how to derive?

Answer 22

yes i do

Answer 23

A = 4pi r - 1/r^2 then correct?

Answer 24

A' that is lol

Answer 25

Yes that is what I just got

Answer 26

if we combine these with like denoms we get:

\[A'=\frac{4\pi r^3 - 1}{r^2}\] correct?

Answer 27

Yes that looks good

Answer 28

\[A'=0=4pir^3 -1\] so what are the zeros?

Answer 29

r = ? the easiest way is to plug this into wolframs site and itll tell us right away ;)

Answer 30

http://www.wolframalpha.com/

Answer 31

okay and the equation that I would type in would be \[\sqrt[3]{1/4\pi}\]

right

Answer 32

4pi r^3 - 1 :) and i got abt .4 as the only root

Answer 33

oh okay you don't have to get r by itself

Answer 34

not with wolframs site; itll do it for you

Answer 35

r = aprrox. .430127

Answer 36

okay and r^2 is the diameter and I can use that to solve for Amin

Answer 37

Yes; or simply use it to solve 'h'

Answer 38

Yeah I guess that would be easier lol

Answer 39

how many cups to a quart? 8 or 4?

Answer 40

4 cups

Answer 41

4 cups to 1 quart and we need half a cup

so 1/8th of a quart right?

Answer 42

yes

Answer 43

1/2 cup = 946/8 = 118.25 centimeters squared (cc) then

Answer 44

2 = abt 2.1 centimeters then

Answer 45

r = abt. 2.1 centimeters

Answer 46

A = 2pi(2.1)^2 + 1/(2.1)

Answer 47

Total min area = 28.18 cc

Answer 48

wouldn't it be 2.1^4 since it was d^2

Answer 49

oh never mind I see it now

Answer 50

diameter = 2r = 2(2.1) = 4.2

V = 128.25 = 2pi(2.1)^2 * h

128.25
---------- = h = abt. 4.63
2pi (2.1)^2

Answer 51

Thank you very much I've been working on this problem for about 3 hours and you figured it out in 10 minutes lol! You're a lifesaver!

Answer 52

is it right? cuase i might see an error in it; i allowed the area of the material to be 2 circles, top and bottom of a can; and a height...

Answer 53

Yes it is it matches up to the parts that I had figured out

Answer 54

cool :)

Answer 55

Thanks again!