Hi. I have these math puzzles for homework and I was hoping you could help me a little with these.
1) You have a round cake and 150 guests. What is the minimum number of slices completely through the cake (does not need to be through the center) that you must make to guarantee each person has a piece (they don't need to be equal sized.)?
2) There is a goat tied to the corner of a rectangular barn on a 30 foot lead and the barn is 20 feet by 20 feet (floor), what is the maximum grazing area? Remember the rope can't go all the way around the barn.
Please give me a hint. You don't have to solve, but some clues would be nice. For the first problem, apparantly there's supposed to be an equation, but my teacher wanted us to try to solve it ourselves.
Thanks so much!
I have tried these both, but I could not get a reasonable answer. I'm now confused…
Hi,
1) You have a round cake and 150 guests. What is the minimum number of slices completely through the cake (does not need to be through the center) that you must make to guarantee each person has a piece (they don't need to be equal sized.)?
If the cake is cut once, there are 2 pieces. If it is cut again so that the second cut crosses the first cut, there are 4 pieces. If it is cut again so that the third cut crosses both of the previous cuts, there are 7 pieces. This pattern continues.
The cake started as one piece. Do you see the first cut added one more piece, the second cut added 2 more pieces, the third cut added 3, the fourth cut added 4 more, etc.
If you continue this, you will see it takes 17 vertical cuts to have 150 pieces of cake cut.
There need to be 17 vertical cuts. <==ANSWER
The formula for this is y = ½x² + ½x + 1. <==ANSWER
** I'm sure you don't want to do this, but if you made 12 vertical cuts that cross each other, and then made a horizontal cut through the cake, there would be more than 150 pieces of cake.
2) There is a goat tied to the corner of a rectangular barn on a 30 foot lead and the barn is 20 feet by 20 feet (floor), what is the maximum grazing area? Remember the rope can't go all the way around the barn.
3/4 of a circle with radius of 30 could be formed by the goat, In addition, 1/4 of a circle of radius 10 could be extended past the two sides of the barn.
3/4 x π(30)² + 2(1/4) x π10² = 675π + 50π = 725π <==ANSWER
I hope that helps!! 
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for 1, im guessing 75 cuts. because if u are serving 2 ppl, u only need 1 slice. if ur serving 4, u only need 2. if ur serving 8, u only need 4. so i guess u gotta divide 2. so u only need 75 cuts to give each person a slice.
2.the area of the whole barn is 400 feet. and, idk wat to do after. haha.
References :
For number 1:
The absolute best you can do per slice is to double the number of pieces of cake. (That is, you cut every piece in half. You would probably have to rearrange the pieces after each cut).
So the number of pieces after n cuts would be 2^n.
Thus, the minimum n would be 8 to get 150 pieces of cake. 7 slices would only get you 128 pieces.
Number 2 is significantly more complicated than rhino thinks. You have to take into account the fact that the rope cannot go through the barn. You would have 3/4 of a circle with the rope's radius, but when the goat gets to the corner, it is like he has a new, shorter rope anchored there.
References :
well for the first time each time you make a cut all the way through the cake u divide the cake in half.
To find the cuts needed find n
150 = 2^n
Find the value of n that makes it greater than 150 and n must be a whole number so round up
With the second one find the circumfrance and subtract the area of the barn. It may help to include a drawing
Good Luck
References :
1) What I like to do for these problems is to work with a smaller number (of guests) and figure out how it works — see if I can find a pattern.
I'll make columns with
guests cuts
2 ——– 1
3 ——– 2
4 ——— 3
Ah — so it looks like I make one less cut than the number of guests.
(After seeing other responses, I see I made an assumption that the cuts were to be non-intersecting.)
If the lines can intersect each other this could get messy.
Say you put 4 lines through the center point resulting in 8 pieces.
Now you could put one line parallel to one of those lines and get 12 pieces.
Actually I think the minimum number would be found with parallel and perpendicular lines. Easier to think about in terms of a rectangle.
15 * 10 = 150 sq units
Breaking these down into their prime factors
3 * 5 * 2 * 5 = 150 sq units
6 * 25 = 150 sq units
3 * 50 = 150 sq units
What is the smallest sum:
15 + 10 = 25
6 + 25 = 31
3 + 50 = 53
So 15 cuts one way and 10 the other for a mimum number ot 25.
Ah — so get the prime factors for number of guests and find the least sum of two products.
2) Here I drew a picture — break the problem down into its parts.
He can go 3/4 of a circle of radius 30 ft. The barn will block 1/4 of the circle. Additionally at the corners of the barn opposite where he is tied, he can go a quarter of a circle of radius 10 ft (30 lead – 20 ft barn length). There will 2 of these areas at each opposite corner.
Area of a circle = pi times radius sqaured
So the area the goat can graze (Goat Grazing Area)
GGA = (3/4 pi 30^2) + 2 ( 1/4 pi 10^2) sq ft.
References :
Hi,
1) You have a round cake and 150 guests. What is the minimum number of slices completely through the cake (does not need to be through the center) that you must make to guarantee each person has a piece (they don't need to be equal sized.)?
If the cake is cut once, there are 2 pieces. If it is cut again so that the second cut crosses the first cut, there are 4 pieces. If it is cut again so that the third cut crosses both of the previous cuts, there are 7 pieces. This pattern continues.
The cake started as one piece. Do you see the first cut added one more piece, the second cut added 2 more pieces, the third cut added 3, the fourth cut added 4 more, etc.
If you continue this, you will see it takes 17 vertical cuts to have 150 pieces of cake cut.
There need to be 17 vertical cuts. <==ANSWER
The formula for this is y = ½x² + ½x + 1. <==ANSWER
** I'm sure you don't want to do this, but if you made 12 vertical cuts that cross each other, and then made a horizontal cut through the cake, there would be more than 150 pieces of cake.
2) There is a goat tied to the corner of a rectangular barn on a 30 foot lead and the barn is 20 feet by 20 feet (floor), what is the maximum grazing area? Remember the rope can't go all the way around the barn.
3/4 of a circle with radius of 30 could be formed by the goat, In addition, 1/4 of a circle of radius 10 could be extended past the two sides of the barn.
3/4 x π(30)² + 2(1/4) x π10² = 675π + 50π = 725π <==ANSWER
I hope that helps!!
References :