Question

Simple math question: Can my surface area change if my volume stays the same?

Answer 1

yes.

Answer 2

explain plz

Answer 3

The volume of a sphere = 4/3 * pi * r^3
The volume of a cube has = x^3 where x = length of an edge.
if you set these to be = lets say 1cm^3 as teh area and solve for x and r each

Answer 4

we have 4/3*pi*r^3 = 1 means r ~ 0.62
and we have x^3 = 1 means x = 1

Answer 5

the surface area of a cube = 6 * x^2 = 6
The surface area of a sphere = 4pi * r^2 = 4.83

Answer 6

So my problem says, a can has the volume of 200, and i am to find the dimensions of said can, so that they are the most cost effective (smallest)

Answer 7

you want to minimize surface area while maximizing volume.

Answer 8

yes but the volume must stay at 200

Answer 9

V = pi*r^2 * h
SA = 2pi*rh + 2pi*r^2

Answer 10

yes

Answer 11

but the dimensions must be altered so i find the ideal surface area

Answer 12

One second Let me work on it on paper first.

Answer 13

What math are you in algebra or calc?

Answer 14

ok, i was thinking of just plugging in functions until i find the answer out.

Answer 15

If you solve volume for h, you get h = 200/(pi*r^2)

Answer 16

plug this into the h in the SA equation to get :
\[SA = 2\pi r(200/(\pi*r^2))+ 2\pi*r\]

Answer 17

Now, what you want to do is solve this equation for a value of r that gives the smallest SA

Answer 18

so which equation a i solving first??

Answer 19

you solve volume for the variable h. then you plug in what h equals into the h in the surface area equation

Answer 20

ohhhhhhh
makes sense

Answer 21

wait, so do i plug in 200 as SA?? @manumben

Answer 22

Which math class are you in? calculus?

Answer 23

Honors Geometry

Answer 24

ok then im helping you the wrong way... one moment

Answer 25

lol ok

Answer 26

h = 2r is the most efficient dimensions for a can

Answer 27

this is because it is closest to a sphere which has the greatest surface area to volume ratio

Answer 28

200 = pi * r^2 * h
substitute using h= 2r or r = 1/2 h

Answer 29

so then as long as my height is twice as big as my length and the volume multiplies to 200 im good

Answer 30

yup

Answer 31

Noice!

Answer 32

but the surface area of a 10X20 can is 1668, so wouldnt it be more efficient to use a can with a radius of 5 and height of 3??

Answer 33

@manumben

Answer 34

@manumben