Question

A design on the surface of a balloon is 5 cm wide when the balloon holds 71 cm^3 of air. How much air does the balloon hold when the design is 10 cm wide? Explain the method you use to find the amount of air.

Answer 1

I believe this is the volume of a sphere formula, are you familiar with it?

Answer 2

not really

Answer 3

Hmm... this isn't super straightforward. I think first things first, find the radius.I have an idea how to do this, but it seems like a fair bit of work, maybe there's an easier way. \[\Large V = \frac{ 4 }{ 3 } \pi r^3\]you're given volume = 71 so plug that in, find r.

Answer 4

\[71 cm^3=\frac{ 4 }{ 3 }\pi r^2\]

Answer 5

thats as far as i got....

Answer 6

ok let me do that

Answer 7

@satellite73 maybe you or someone else can think of an easier way to solve this than what i have in mind :/ maybe we can use the surface area, instead of needing things like arc length.

Answer 8

Maybe we can use the proportion of surface area to volume to make this easier...

Answer 9

\[71*3=\frac{ 4 }{ 3 }\pi r ^2*3\]
\[\frac{ 123 }{ 4 }=\frac{ 4\pi r^2 }{ 4 }\]
\[\frac{ 30.75 }{ \pi }=\frac{ \pi r^2 }{ \pi }\]
\[9.788=r^2\]
am i on the right track

Answer 10

I think so, but maybe we don't even need the radius... but i guess it might come in handy. maybe we can use the fact that the surface area increases by a factor (of 4 if i'm thinking about it right) so the volume must also increase by a constant factor - which would make this much simpler. @terenzreignz

Answer 11

Like if the surface area quadruples (which i *think* it does based on the design radius doubling...), that'd mean the radius must have doubled - which would mean the volume must be octuple what it was. I'm just not sure this is actually legit.

Answer 12

@e.mccormick

Answer 13

@shamil98

Answer 14

@thomaster @radar

Answer 15

is this some kind of proportionality problem?

(5/10)^3 = 71/V, and solve for V

Answer 16

i dont think so not sure to be honest

Answer 17

V = 568, a whole number, so it must be right XD

But then still, that is based on the assumption that the length of the design grows proportionally as the balloon gets bigger. Of course, i'm not sure if that is actually true in real life

Answer 18

\(S\cdot\dfrac{r}{3}=V\)

Answer 19

@e.mccormick can you explain a little @sourwing im lost on how you got 568

Answer 20

And you can find the radius from the first volume. Since your looking at just the width you are only changing in one dimension.

Answer 21

(5/10)^3 = 71/V
V(5/10)^3 = 71
V = 71 / (5/10)^3 = 568

Answer 22

(5/10)^3 would i need to reduce that?

Answer 23

if you want to. The final number is the same anyway

Answer 24

Yep, and it checks against doing it with more steps.