Question

Does a larger surface area always mean there is a larger volume?

Answer 1

not always think of a tall skinny glass compared to a short fat with equal volumes one the tall skinny one has more surface area but less volume the fat one will have less surface area but the same volume.

Answer 2

Which problem is possible to solve? (Hint: Don't calculate numbers, think theoretically)

-Find a cylinder box of volume 3 L with the 1) smallest or 2) largest surface area possible?

Is 1 or 2 possible to solve? (One of them is impossible)

Answer 3

I wanna say 2 but Im not quite sure I understand what the question is asking

Answer 4

One of the problems is not possible to solve. Try to pick the question that is possible and explain why.

Answer 5

Not necessarily

Consider the case of a prism (polygon faced extruded)
If all we change is the base face, then the surface area will increase with the perimeter of the face, and the volume will increase with the area of the face.

Imagine the face is square (in black), and we change the face to the wacky green shape (with larger perimeter, but smaller volume). Well, what we have done to the prism is increase its surface area, but decrease its volume.

Answer 6

@apple_pi
Do you know which problem is possible to solve? (Hint: Don't calculate numbers, think theoretically)

-Find a cylinder box of volume 3 L with the 1) smallest or 2) largest surface area possible?

Is 1 or 2 possible to solve? (One of them is impossible)

Answer 7

hang on

Answer 8

I only asked the 1st question because of this question!:

Which problem is possible to solve? (Hint: Don't calculate numbers, think theoretically)

-Find a cylinder box of volume 3 L with the 1) smallest or 2) largest surface area possible?

Is 1 or 2 possible to solve? (One of them is impossible)

Answer 9

In regards to your problems, consider an extreme case...

We could have an extremely thin cylinder that is (almost) infinitely long.
This would have a whopping surface area, despite it being incredibly thin.

Now consider the other extreme case, with an extremely wide circle. The wider the circle, the greater the surface area (the curved surface area becomes negligible)

So therefore you would have to look for a minimum surface area.

Answer 10

@apple_pi But the volume has to be 3L, so how can you keep increasing the surface area? Wouldn't there have to be a maximum?

Answer 11

il explain that :)

V = pir^2*h = 3
SA = pir^2 + 2pir h
= pir^2 + 2*3/r

keeping V = 3,
as r-> infinity, SA -> infinity


so, u can choose any arbitrarily large value for r, and u can get a larger surface area always. so there is no fixed box wid largest surface area.

Answer 12

woww ok I get it!,

What if it was a rectangular box and not cylinder?

Answer 13

apply same logic :)

Answer 14

V = LBH = 3
SA = 2*LB + 2(L+B)H
= 2*LB + 2(L+B)3/LB
= 2(LB^2 + 3L + 3B)/LB

keeping V = 3,
as L or B-> infinity, SA -> infinity


so, u can choose any arbitrarily large value for L or B, and u can get a larger surface area always. so there is no fixed box wid largest surface area.

Answer 15

but when L or B becomes larger, H becomes smaller, so doesn't it cancel out the effect?

Answer 16

for a sphere its different : larger surface area always means a larger volume

Answer 17

very good question :)

look at below equation :-
SA = 2*LB + 2(L+B)H

Answer 18

as L-> infinity, we dont even have to bother about second term.
first term, the base area overtakes the reduction in area due to decrease in height

Answer 19

wait, so LWH, if L gets bigger, W and H get smaller.

SA=2LW + 2LH + 2 WH
so if L gets bigger, but W gets smaller, isn't LW offset?

Answer 20

oh ya, let me think W also depends on L

Answer 21

Consider this :-
Keep W fixed.
Increase L,
Decrease H.

so that volume stays fixed

Answer 22

say W = 1 ?

SA = 2*LW + 2(L+W)H
= 2*L + 2(L+1)3/L
= 2*L + 2 + 6/L

Answer 23

as u take very large values for L, SA increases eventhough H is decreasing

Answer 24

the effect of 6/L wont matter when u take very large values for L so that :-

change in (2L) > change in (6/L)

Answer 25

wait but H =3/LW, so as L increases it's 2L + 2LH +2H, so isn't LH offset and 2L offset by 2H?

Answer 26

change everything into L's and analyze what happens as u increase L

Answer 27

H = 3/L right ?

Answer 28

substitute that in ur SA expression

Answer 29

Wait H=3/LW!

Answer 30

we fixed W = 1 !

2L + 2LH +2H

put H = 3/L

Answer 31

2L+6 + 6/L

Then when L gets bigger, the SA gets bigger!