I was buying birthday pastry this week and saw that the 11" x 3" lemon pie is $24 while the 4" diameter round version of this pie is $10. And you know what is the best part of those pies? The buttery crust perimeter that is touching the delicious tart lemon filling and heavenly soft meringue. If you want to spend $50 on the pies and would like to have the maximum perimeter-to-cake size ratio, what pies should you pick?
10 comments:
The perimeter of a circle is given by:
C = 2 π r
For the given pie, that will be:
C = 2 * π * 2 = 12.57
The perimeter of a rectangle is given by:
P = 2 (l + w)
For the given pie, that will be:
P = 2 (11 + 3) = 28
You want to spend a maximum of $50. Assume that you can only purchase whole pies. The rectangle pie is $24, the circle pie is $10. Looking at the prices, I note that if you buy no circles, the maximum number of rectangle pies is 2. Start there, reduce rectangle pies by one until zero, and calculate the maximum number of circle pies:
#Rect = 2
RCost = 48
#Cir = 0
CCost = 0
Total Cost = 48
#Rect = 1
RCost = 24
#Cir = 2
CCost = 20
Total Cost = 44
#Rect = 0
RCost = 0
#Cir = 5
CCost = 50
Total Cost = 50
Calculate total perimeter based on number of circles and rectangles:
2 * (28) = 56
1 * (28) + 2 * (12.57) = 53.14
0 * (28) + 5 * (12.57) = 62.85
Therefore, to maximize the amount of "crust" while spending $50 or less, you should buy 5 circle pies of 4 inch diameter at $10 each.
The perimeter of the rectangle is 11*2 + 3*2 =28
The perimeter of the circle is 2 pi r= 4 pi so this perimeter is around 13 which is inferior to 28
I will buy the rectangle ones.
The pies call me back!
I would choose the circular pie because it's C = 12.56 and it's A = 12.56. This is a ratio of crust to filling of 1. The rectangular pie's P = 28 and it's A = 33. This is a ratio of crust to filling of less than 1. (28/33)
I'm not sure I know what "cake size" means. Is that another term for area? I'll take it so. If it is not, and I think of something else, I'll add it later.
Perimeter of rectangle = (3+11)*2 = 28"
Circumference of circle = 2*pi*r = 12.57
Area of rectangle = 3*11 = 33
Area of Circle = pi*r^2 = 12.57
Let the number of rectangles = x
Let the number of circular pies = y
P/Area = (28*x + 12.57*y)/(33x + 12.57*y)
The maximum perimeter to area ratio occurs when there are no rectangles and 5 circular pies. Any other combination produces a smaller perimeter to area ratio due to the smaller perimeter total (using the rectangular pie) to area ratio. 28 is smaller than 33
5 round pies
For $50. you could buy 2 rect. pies or 5 round pies.
For the two rect. pies:
Perimeter = (11 +3) x 4 = 56"
Area = (11 x 3) x 2 = 66 sq. "
Ratio = 56/66 or 28/33
For five circular pies:
Circumference = (pi)d = (3.14) x 4 = 12.56 x 5 = 62.8
Area = (pi)r2 = (3.14) x 4 = 12.56 x 5 = 62.8
Ratio = 62.8/62.8
You will get more crust per pie with the round pies.
I've been assuming that both pies are round -- with the bigger one having a 11" diameter and 3" depth, and the smaller one having a 4" diameter and 3" depth, but your graphic with the puzzle (with I just saw on your puzzle page now, not in your email) suggests otherwise. Is the larger pie indeed a rectangle?
Thanks,
TracyZ
I originally was thinking of the first pie as being circular since most pies are and since my email program didn't show me the schematic of the two pies. I was also thinking about the total amount of crust, not just the top perimeter. Now I understand that the problem better. :)
I am still not sure whether the phrase "cake size" is referring to the top surface area or total pie volume of each cake/pie. Regardless, I think the answer is the same.
For the first pie, the rectangular one:
the top crust is 28 inches in length ((11+3)*2) = 28)
the top surface area = 33 sq. inches
the volume = 33 * h cubic inches (where h is the height of the pie)
For the second pie, the circular one:
the top crust is 4*pi inches in length or ~12.57 inches
the top surface area = 4*pi sq. inches
the volume = ~ 4*pi*h (roughly, unless the bottom diameter of the pie plate is significantly less than the top diameter).
with x = number of first size pies and
y = number of second size pies,
this problem seeks to maximize the perimeter-to-cake- size ratio, subject to the following constraints:
$24x + $10y <= $50
x and y must be whole numbers
given these constraints, there are three possible (x,y) pairs that could have the max perimeter-to-size ratio:
(0,5), (1,2), and (2,0)
of these options, the best is (0,5).
(0,5) spends all the available money ($50), and has a perimeter-to-surface-area ratio =
4*5*pi inches / 4*5*pi sq. inches =
1 inch per sq. inch
Options (1,2) and (2,0) neither spend all the available money (spending only $44 and $48 respectively) nor have as good a perimeter-to-size ratio.
TracyZ
A little after deadline, but hey, I was busy with Mom's Day celebrations! :-).
To answer the exact question, I will assume that pie depth is the uniform and the same for two kinds of pies and so "cake size" refers to area. With that, the rectangle pie's perimeter to size ratio 11*2+3*2/11*3=28/33. For the round pie = 2*Pi*2/Pi*2^2=1. So to maximize the ratio, get the round pies. Now, if all you care about is the crust, then perhaps the question should not involve the area, but the money, i.e. what pies get more perimeter per dollar? Again, round ones win, as for $50 you get 2*Pi*2*5=62.8 for $50, i.e. 1.256 inches perimeter per dollar, whereas with rectangle ones you get 28*2=56 for $48, i.e. 1.1667 inches per dollar. One sure way to get more perimeter is to convince the bakery to change the pies to be even narrower than 3". Maybe 2" by 16" :-).
Happy Mother's Day!
5 round pies!
It is pretty neat that they have a crust-per-area ratio of 1. And therefore when diving them equally between some eaters each will get the same amount of crust.
I guess this is why round pizzas are more popular than square ones.
TracyZ - sorry for confusing you with the square pies. You are right - I added the pic later on and only to the website version.
Ilya - you are right - going for even smaller circles will be yummier. Some bakeries and catering got this right.
A puzzle point for everyone.
Dennis - where is Katrina?
Maria -
She's lurking; she gave birth to our fifth in late December, and between that and keeping up with the other four, she just hasn't found the time to post.... I, on the other hand, tend to work or your puzzles instead of working, so....
:)
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