Math Problem solving- pls help!!!!?

Question

CLASSIC PROBLEM - A traveling salesman (selling shoes) stops at a farm in the Midwest. Before he could knock on the door, he noticed an old truck on fire. He rushed over and pulled a young lady out of the flaming truck. Farmer Brown came out and gratefully thanked the traveling salesman for saving his daughter’s life. Mr. Brown insisted on giving the man an award for his heroism.

So, the salesman said, “If you insist, I do not want much. Get your checkerboard and place one penny on the first square. Then place two pennies on the next square. Then place four pennies on the third square. Continue this until all 64 squares are covered with pennies.” As he’d been saving pennies for over 25 years, Mr. Brown did not consider this much of an award, but soon realized he made a miscalculation on the amount of money involved.
a) How much money expressed in dollars would Mr. Brown have to put on the 32nd square?
Answer:
Show work in this space




b) How much money expressed in dollars would the traveling salesman receive in total if the checkerboard only had 32 squares?
Answer:
Show work in this space




c) Calculate the amount of money necessary to fill the whole checkerboard (64 squares). How money expressed in dollars would the farmer need to give the salesman?
Answer:
Show work in this space

Answer 1

This is a classic problem that deals with exponential functions. On the first square, 2^0 ("two to the zero power") pennies (1), on the second square 2^1 = 2, 3rd square 2^2 = (2)(2) = 4, 4th square 2^3 = (2)(2)(2) =8 pennies etc.

a) on 32nd square, 2^31 = 2 multiplied by itself 31 times. I believe this is 2,147,483,648 cents, but check my arithmetic. To find dollars divide by 100 (move the decimal point 2 places to the left)
Answer: $21,474,836.48

b) to solve this, just add 2^0 + 2^1 +...+ 2^31 (a scientific calculator will help with that. Then divide by 100 to find dollars instread of cents.

c) to solve this, just add 2^0 + 2^1 +...+ 2^63 (a scientific calculator will help with that. Then divide by 100 to find dollars instread of cents.

What a great question, albeit time consuming.

Answer 2

CLASSIC PROBLEM - A traveling salesman (selling shoes) stops at a farm in the Midwest. Before he could knock on the door, he noticed an old truck on fire. He rushed over and pulled a young lady out of the flaming truck. Farmer Brown came out and gratefully thanked the traveling salesman for saving his daughter’s life. Mr. Brown insisted on giving the man an award for his heroism.

So, the salesman said, “If you insist, I do not want much. Get your checkerboard and place one penny on the first square. Then place two pennies on the next square. Then place four pennies on the third square. Continue this until all 64 squares are covered with pennies.” As he’d been saving pennies for over 25 years, Mr. Brown did not consider this much of an award, but soon realized he made a miscalculation on the amount of money involved.
a) How much money expressed in dollars would Mr. Brown have to put on the 32nd square?
Answer:
the amounts on each square are in GP with the first term as 1 and the common ratio as 2.
thus the n-th term will be 2^(n-1)

thus the 32nd square will have 2^31 in pennies.
covert to dollars

b) How much money expressed in dollars would the traveling salesman receive in total if the checkerboard only had 32 squares?
Answer:
that would be (2^32-1)penies..convert to dollars.




c) Calculate the amount of money necessary to fill the whole checkerboard (64 squares). How money expressed in dollars would the farmer need to give the salesman?
Answer:
2^64-1
sum of the GP.

answers in pennies because I'v forgotten the conversion of pennies to dollars...
correction..just found out that they stand for one cent and so just divide the answers by 100 to get the number of dollars..

Answer 3

(a)
square 1 ; 1
square 2 ; 2
square 3 ; 4 = 2²
square 4 ; 8 = 2³
square 5 ; 16 = 2⁴
square 6 ; 32 = 2⁵
:
:
square 32 ; 2^31 = 2,147,483,648 pennies = 21,474,836.48 dollars

Hmmm ? Must have had a strong floor in his house !

(b)
1
1 + 2 = 3 = 2² -1
1 + 2 + 4 = 7 = 2³ - 1
1 + 2 + 4 + 8 = 15 = 2⁴ - 1
1 + 2 + 4 + 8 + 16 = 31 = 2⁵ - 1 and so on

Total is 2^32 -1 = 4294967295 pennies = 42,949,672.95 dollars.

(c) same method total = 18,446,744,073,709,551,616 pennies = 184,467,440,737,095,516.16 dollars

He must have been exhausted just collecting them - and he certainly puts ol' Bill Gatesey to shame wealth-wise !

Answer 4

The general formula is
Sum (from k = m to k = n) of ar^k =
a[r^m - r^(n+1)]/(1-r)

Our formula is
Sum from (k = 1 to k = n) of (1/2)2^k =
(1/2)(2 - 2^(n+1))/(1-2) = -1 + 2^n

so:

a) the 32nd square holds
(1/2)(2^32) = 2147483648 cents
or $21,474,836.48

b) -1 + 2^32 = $42,949,672.95

c) -1 + 2^64 (My calculator won't go that high - you could probably do it in Excel or something.)

Answer 5

a) (2**31) / 100 where ** = to the power of
1st square = 2**0/100 = $0.01
2nd square = 2**1/100 = $0.02
3rd square = 2**2/100 = $0.04
etc
b) sum of all 2**n where 0 <= n <= 31, divided by 100
c) sum of all 2**n where 0 <= n <= 63, divided by 100

Answer 6

a) 2^31/100 = $21474836.48

b) (2^32 - 1)/100 = $42949672.95

c) (2^64 - 1)/100 = very big number