# I need help with the following question. Thank you!

• Jul 3rd 2014, 12:16 PM
Jannylee
I need help with the following question. Thank you!
How many quarters do you need to fill up a US phone booth?

The inside area of the phone booth is 32x32 square inches. Total area is 1024 squre inches.
Dimension of a US quarter:

Diameter 24.26 mm (0.955 in)
Thickness 1.75 mm (0.069 in)
Weight 5.67 gm
• Jul 3rd 2014, 12:39 PM
romsek
Re: I need help with the following question. Thank you!
Quote:

Originally Posted by Jannylee
How many quarters do you need to fill up a US phone booth?

The inside area of the phone booth is 32x32 square inches. Total area is 1024 squre inches.
Dimension of a US quarter:

Diameter 24.26 mm (0.955 in)
Thickness 1.75 mm (0.069 in)
Weight 5.67 gm

you can get a rough estimate by dividing the volume of the phone booth by the volume of a quarter, but that assumes quarters fit perfectly into your booth, which they don't.

You don't give the height of the phone booth so I can't determine the volume but you can get an approximation of the number of quarters per inch height of the booth.

Booth volume = 32*32*1 = 1024 cubic inches volume per inch of height.

The quarter has volume $V_q=\pi (0.955/2)^2*0.069 \approx 0.05$ cubic inches

So the number of quarters that will fit into a 1 inch tall booth is $N\approx \dfrac {1024}{0.05} \approx 20480$ quarters per inch of phone booth height.

A better approximation takes into account the fact that quarters are round.

You will only be able to fit $\left\lfloor \dfrac {32}{0.955}\right\rfloor=33$ quarters in a row in the booth.

So per quarter thickness of phone booth height you have $33^2 = 1089$ quarters.

You can stack $\dfrac 1 {0.069}\approx 14.5$ quarters per inch of phone booth height.

So you can fit approximately $1089 * 14.5 = 15790.5$ quarters in 1 inch of phone booth height. A fair bit less than the rougher estimate above.
• Jul 3rd 2014, 12:44 PM
ebaines
Re: I need help with the following question. Thank you!
You ask about "filling up" a phone booth, but don't mention how hight the ceiling is. So perhaps what you meant to ask is "how many coins does it take to cover the floor." To figure that you need to think about how the coins are arranged. So first - how many coins can fit in the first row? Then for the second row - should the coins be lined up in a square array, so that the coins of the 2nd row are in line with the coins in the first row? Or should the second row be shifted to line up between the coins of the first row - this is a tighter packing method that alloys you to squeeze in more coins. You'll have to check whether a complete row of coins will fit in that shifted manner, then figure out the center-to-center spacing of one row to the next, and from that determine how many rows of coins will fit on the floor of the booth. Try it, and let us know what you find.
• Jul 3rd 2014, 12:50 PM
Jannylee
Re: I need help with the following question. Thank you!
Wow, thanks for the quick response! The interior height of the phone booth is 89.028 inches.

Thank you!
• Jul 4th 2014, 03:29 PM
HallsofIvy
Re: I need help with the following question. Thank you!
You cannot cover the floor of the phone both with quarters because you cannot "tile" the rectangle with circles. The best you can do is treat a single stack of quarters as a square object with sides .955 inches long and so area $.955^2$ square inches. Divide the 1024 square inch area of the floor by that to see how many stacks of quarters you can have. And divide the height of the phone booth by .069 to find how many quarters there would be in each stack. Since the number of stacks and the number of quarters in a stack must be integers, drop any fractional part (do NOT round up).