A cylindrical cardboard mailing tube has a diameter of 90 mm. and a height of 600 mm. A total of 20% of the tube’s capacity is filled with spherical packing foam. Each foam sphere has a radius of 15 mm.

Approximately how many pieces of foam are in the tube?
Use 3.14 to approximate pi and express your answer as a whole number.

Question

A cylindrical cardboard mailing tube has a diameter of 90 mm. and a height of 600 mm. A total of 20% of the tube’s capacity is filled with spherical packing foam. Each foam sphere has a radius of 15 mm.
 
Approximately how many pieces of foam are in the tube?
Use 3.14 to approximate pi and express your answer as a whole number.

mathstudent55 · Answer

The best you can come up with is an approximation of the maximum that could possibly fit since the foam spheres leave gaps between them.

anonymous · Answer

|dw:1428414685597:dw| a drawing should help ;)

mathstudent55 · Answer

First, find the volume of the cylinder using the formula for the volume of a cylinder,
$V = \pr r^2 h$
Notice you need the radius in the formula and you are given the diameter.

anonymous · Answer

Math seems to know what he's doing, so I'll leave it to him, good luck!

mathstudent55 · Answer

Then take 20% of the volume of the cylinder.
Now find the volume of a sphere using the formula
$V  = \dfrac{4}{3} \pi r^3$
Then divide the 20% of the volume of the cylinder by the volume of one sphere.
That will give you the highest possible number of spheres that could fit if there were no gaps between spheres, but since there are gaps, the actual number will be lower.

anonymous · Answer

wait so do i divide 90/2 to get the radius and then i do $$V = \pr r^{2}h $$ @mathsciencehistory  ??

mathstudent55 · Answer

Yes. Correct. The radius is (90 mm)/2 = 45 mm

anonymous · Answer

and then what do i do? @mathstudent55

mathstudent55 · Answer

You found the volume of the cylinder?

anonymous · Answer

3.82×106 thats what i got @mathstudent55

mathstudent55 · Answer

Great.
Now take 20% of that volume.
That is the part of the cylinder that has foam spheres in it.

anonymous · Answer

so do i multiply 3.82 x 106 and then take 20% ?

mathstudent55 · Answer

Multiply 3.82 x 10^6 by 0.2 since 0.2 = 20%

anonymous · Answer

i got 19100000 @mathstudent55

mathstudent55 · Answer

Did you multiply by 0.2 or divide by 0.2?

anonymous · Answer

divide

mathstudent55 · Answer

You need to multiply, not divide.
3.82 x 10^6 * 0.2 = 763,000

anonymous · Answer

ohh

mathstudent55 · Answer

This new number, 763,000 mm^3 is the part of the cylinder occupied by foam spheres. To find how many spheres can fit there, we need to find the volume of one sphere.
I wrote above the formula for the volume of a sphere. Use the radius of the spheres and find the volume of one sphere.

anonymous · Answer

14137.17 thats the volume for one sphere @mathstudent55

mathstudent55 · Answer

Correct.
Now divide the volume of the 20% of the cylinder by the volume of one sphere.
What is 763,000/14130 = ?

anonymous · Answer

53.9985etc @mathstudent55

mathstudent55 · Answer

Correct, so the largest possible number is approximately 54 spheres.