A community was experiencing a serious problem, The King of that community calle

Question

A community was experiencing a serious problem, The King of that community called upon all his advisors to solve the problem but non of them could. so he called upon all the people of the community decreeing that if anyone could solve the problem he would grant that person one wish, as long as the wish was not the throne or his head. Many people tried to solve the problem, but failed. Finally a beggar in the Kingdom came forward and solve the problem. The King asked the beggar to state his wish. the beggar stated his wish as follows:

Give me bags of rice based on the squares of a chess board. Starting with the 1st square, give me one bag then double the number bags as you proceed along till you get to the 64th and final square.

SOLVE THE FOLLOWING PROBLEMS BASED ON THE BEGGAR’S WISH

5. Construct a table of values for the bags of rice for at least the 1st six squares.

6. Use your table of values to graph the function for the bags of rice on a square

7. If the beggar decided to sell the bags of rice at $3.00 per bag;

(a) How much did he get for selling the bags on the 15th square.

(b) How much did he get for selling every bag from the 1st to the 15th square.

(c) What is the function for the sale of the bags of rice on each square.

8. If the beggar decided to give every bag of rice on the 12th square to charity, how much did he make if he sold every bag from square 1 to 15 minus what he gave to charity.

9. What percent of his earnings did he give to charity.

10. If the beggar decided to sell the bags of rice at $5.00 per bag;

(a) How much did he get for selling the bags on the 15th square

(b) How much did he get for selling every bag from the 1st to the 15th square.

(c) What is the function for the sale of the bags of rice on each square.

11. How much did the beggar make if he sold all the bags on 64th square

(a) At the $3.00 rate.

(b) At the $5.00 rate.

12. Graph the function for the sale of the bags of rice on each square;

(a) At the $3.00 rate.

(b) At the $5.00 rate.

Explanation / Answer

5) Let the square count be n and no of bags in that square be f(n)

Note above pattern forms a gp as every term is double of the first. So r = 2

Nth term of gp = a1rN-1 where a1 is first term of gp.

6) Function for above table can be

f(n) = 2 n-1

check :

f(1) = 1

f(2) = 2

f(3) = 4

and so on..

Note formula for sum of Finite G.P. till nth term = a1(rn - 1)/ (r-1)

Nth term of gp = a1rN-1

7 )S.P.for each bag = $ 3

a) Now no. of bags form a G.P. which is to be calculated till 15th term, 1,2,4,8,.... here r = 2

no of bags on 15 square = f(15) = 1(215-1 ) = 214 = 16384

so S.P. for bags on 15th square = 16384 x 3 = $49,152

b) No. of bags from 1st to 15th

i.e. Sum of gp till 15th term =

1 + 2+ 4+ 8 +....+16384 = 1 (215 - 1)/ (2-1) = 32768 - 1 = 32767

S.P. = 32767 x 3 = $98,301

C)Function for No. of bags sold from 1st to Nth square = 1(2N -1)/ (2-1) = 2N -1

8) No of bags on 12th square = 212-1 = 211 = 2048

Cost of these bags = 2048 x 3 = $ 6,144

Money he made after giving bags of square 12 to charity = Money made from bags sold till 15th square - charity amount = 98,301 - 6144 = $92,157

Square count (n) bags of rice. (f(n)) 1 1 2 2 3 4 4 08 5 16 6 32

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