## Unlimited Crown Candy Ice Cream for Under $6.50

August 11, 2007 at 9:43 pm | Posted in Gabe Bullard | 4 CommentsTags: food, math

While at Crown Candy with some friends, I noticed a strange pattern with the prices for ice cream.

One pint of ice cream cost four dollars.

One quart (two pints) of ice cream cost five dollars

One half gallon (two quarts) of ice cream cost five dollars and fifty cents.

So, as the amount of ice cream doubled, the increase in price halved.

It could be represented by this equation, where y is the amount of ice cream and x is the increase in price:

x=1/y

That’s kind of simple, but for this instance, it’ll work. See, if you wanted twice as much ice cream, the increase, x, would be 1/2.

Using this rough math, we can try to find the limit of x as y approaches infinity (how much will the price go up as you got closer to ordering an infinite amount of ice cream).

The result?

Lim(x) y–>infinity = 0

So while the increase will never become zero, it will get infinitely closer to zero as the amount of ice cream you want gets infinitely. So, theoretically, a little more than $6 should get you all the ice cream in Crown Candy.

Note: They won’t really give you unlimited ice cream, no matter how mathematically sound your argument is.

But, if somehow the good folks at Crown Candy agree to stick to their formula, they will probably insist on you ordering a finite amount of ice cream. To figure out the exact price, use an exponential decay equation. The first person to do that correctly and tell me the theoretical price of 25 pints of ice cream will get one free pint of Crown Candy ice cream.

Don’t forget to show your work.

## 4 Comments

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Gabe, wouldn’t it be just under $6? Theoretically, if the increase halves each time, starting at $1, it can never pass a total of a $2 increase. Mathematically, I used Pert and integrated over the the range. Unfortunately, I don’t know how to digitize an integration symbol, so I’ll explain that part verbally.

Start with F(t) = Pe^(rt)

F(t) = 1*e^(-.5*t)

Integrate that from t = 0 to 24. Don’t go to 25, since there are only 24 increases (the price for one pint is $4, two is $5, so the 25th pint is the 24th increase).

You end up with 1.999988, which, added to your $4 starting value and rounded to the nearest cent, comes out to $6 flat.

Comment by Patrick (Vacek)— August 12, 2007 #

well patrick, while technically correct, let us keep in mind that the center of a black hole is theoretically infinetely dense. of course we know that it only approaches an infinite density, as though an infinetely dense object would collapse the entire universe (probably). That being said, let’s not round

oh, and ROFL!

Comment by Matthew— August 12, 2007 #

Tax, Patrick. Tax brings it over $6. But you are correct, the price will get infinitely closer to 6 without ever reaching it. Congrats on being the winner.

Comment by Gabe— August 12, 2007 #

Heh, thanks. You guys rule. I did forget about tax, though.

Comment by Patrick (Vacek)— August 12, 2007 #