Need Help with problem involving surface area:

[JenGarcia1122]

Hello,

If someone could please help me with the step by step solution to the following problem that illustrates the ability of packing an enourmous surface area into a very small space.

Problem:
Suppose you had a sphere 12 inches in diameter and you filled it with smaller spheres 2 inches in diameter (hint: about 75% of large sphere's volume will be used if packing is optimal). What is the ratio of the large sphere's surface area to the combined surface area of the small speheres. (Assume spheres are thin).


Thanks for the help!

 

[JenGarcia1122]

The solution I came up with with Equations

Do you know the volume and area formulas?

Is this homework?

Hello Denis,

Thank you for replying... To answer your question this is actually work. I'm a graduate research assistant in public health and I was asked to solve this and to verify that it's correct.. I'm Biology and Chemistry background... and in my Masters degree becoming Biostatistics background.

I haven't seen geometry for a while, so I wanted to make sure this is correct.

I have a feeling part of is wrong because I did not include the small radii area for a single small sphere into my calculation and I believe I should. Below is the solution I came up with, with equations:

Equations~

sphere volume: V=4/3πr³
Diameter=2r
sphere surface area: A=4πr²
conversion equation for spheres from volume to area for spheres: A=π^1/3 (6V)^2/3

My solution ( don't know if it's completely correct):

Large sphere surface area:

D=2r
12=2r
12/2=r
6=r
A=4πr²
A=4π(6)<sup>2

A= 452.399in2

Large sphere volume:</sup>
V=4/3πr³
V=4/3π(6)³
V=904.779 in3

Aproximate volume of small spheres using Large sphere volume:
75% of Large sphere volume:
904.779in3*0.75= 678.584in3 " this volume will be used in conversion equation for small spheres"

Small sphere area and volume:
D=2r
2=2r
1=r
A=4πr² = 4π(1)² =12.566in2 "calculated this area, but don't know if I need to use it or not"~ was not used in this calculation
" did not calculate volume of small sphere, but don't know if I need to use it or not~ was not used in this calculation

Conversion equation from volume to area for small spheres, solving for Area:
A=π^1/3 (6V)^2/3
A=π^1/3 (6*678.584in3)^2/3
A=373.44, "this is the small surface area, accounting for 75% of large sphere surface area,used this value when completing the large sphere to small sphere ratio"

Large sphere to small sphere surface area ratio:
Surface Area Ratio= Large sphere surface area/Small sphere surface area= 452.389in2 "large surface area"/373.44in2 " approximated small surface area, 75% of large sphere surface area"=1.2 " this would be the answer"


*****For curiosity calculated number of spheres:
Approximate surface area of all small spheres/ surface area of one small sphere= 373.44in2/12.5662=29.71 small spheres~ about 30 small spheres.




Thank you.