Question

The first challenge to complete is the Penny Problem. The radio station is giving the winner of this challenge a prize pack that includes tickets to see his or her favorite band in concert. To start off the challenge, the radio station has placed pennies in a cylindrical glass jar. Each penny is 0.75 inches in diameter and 0.061 inches thick. If the cylindrical glass jar containing the pennies has a diameter of 6 inches and a height of 11.5 inches, how many pennies can fit inside the jar? You must show all work to receive credit.

Answer 1

You can calculate the theoretical maximum assuming the volume of the cylinder is entirely occupied by the pennies with no empty spaces left. This is done by dividing the volume of the cylinder by the volume of each penny.

If the above assumption cannot be made, the problem is difficult to solve.

Answer 2

How do i find the volume of the cylinder ?

Answer 3

Volume of cylinder = \(\pi r^2h\), r is the radius and h is the height of the cylinder.

Answer 4

A penny is also a small cylinder and the same formula can be used to calculate the volume of the penny too.

Answer 5

the volume of the cylinder is 108.33 ?

Answer 6

Theoretical maximum number of pennies the jar can hold =
Volume of cylindrical glass jar / Volume of each penny =
\(\Large \frac {\pi R^2 H}{\pi r^2 h} = \frac{R^2H}{r^2h} =
\frac{3^2*11.5}{0.375^2 * 0.061} = ?\)

Answer 7

what about pi ?

Answer 8

so 108.33 is wrong ?

Answer 9

The pi cancels out in the ratio.

But if you want to calculate the volume of the cylinder and then divide by the volume of the penny we can do that too.
Diameter of the cylinder = 6 inches.
Radius of the cylinder = Diameter / 2 = 6 / 2 = 3.
Height of the cylinder = 11.5 inches.
Volume of the cylinder = \(\pi R^2H = \pi * 3^2 * 11.5 = 325.15~ in^3\).

Answer 10

Next, calculate the volume of each penny.

Answer 11

im not getting 325.15

Answer 12

Show your work so I can figure out why.

Answer 13

pi = 3.14
r = 3
h = 11.5

3.14 * 3 ^ 2 = 3.14 * 9 = 28.26

28.26 * 11.5 = 324.99

Answer 14

Well, it is the rounding off error. Use the pi button in the calculator instead of 3.14

Answer 15

for pi i have always used 3.14 ..

Answer 16

do i use the full number then ? 3.14159265359

Answer 17

Yes, that is fine but I am just explaining why I got 325.15 and you got 324.99.

Answer 18

oh okay.... well ill use 3.14 since i always have used it.. so ill leave it at 325.15

Answer 19

In your calculator just punch in: pi x 9 x 11.5 =

Answer 20

Next, calculate the volume of each penny.

Answer 21

Diameter of the penny = 0.75 inches.
Radius of the penny = Diameter / 2 = 0.75 / 2 = 0.375 in.
Height of the penny = 0.061 in.
Volume of the penny = \(\pi r^2h = \pi * 0.375^2 * 0.061 = ?\)

Theoretical maximum number of pennies in the jar = divide volume of cylinder by volume of each penny.

In practice, the maximum number of pennies in the jar will be LESS THAN what the above calculation yields.

Answer 22

.0085 ?

Answer 23

@aum

Answer 24

@aum

Answer 25

No, that is not what I am getting.

Answer 26

@aum what did u get then ?

Answer 27

Volume of the penny = \(\pi r^2h = \pi * 0.375^2 * 0.061 = 0.0269~ in^3\)

Answer 28

ohh

Answer 29

thank you

Answer 30

Theoretical maximum number of pennies in the jar =
volume of cylinder / volume of each penny = 325.15 / .0269 = 12,087 pennies.
The above calculation assumes no empty spaces in the jar. It is like if all the pennies were melted and poured into a jar then the jar can hold 12,087 pennies.

But in practice when a jar is filled with solid pennies there will be lots of empty space in between and so the number of pennies will be lot less than 12,087 pennies.