1.3+Making+a+New+Offer

__**1.3 Making a New Offer**__ When the king told the queen about the reward he had promised the peasant, the queen said, "you have promised her more money than we have in the entire royal treaury! You must convince her to accept a different reward." After much contemplation. the king thought of a plan. He would create a new board with only 16 squares. He would place 1 ruba on the first square, 3 on the next, 9 on the next and so on. Each square would have three times as many rubas as the previous square. A. In the table below, plan 1 is the reward requested by the peasnt and plan 2 the king's new plan. Copy and complete the table to show the number of rubas on squares 1 to 16 for each plan. B. How is the pattern of change in the number of rubas under plan 2 similar to and different from the pattern of change in the number of rubas under plan 1? Similar- Both plan 1 and 2 increase by the power of X. Difference - In Plan 1, it increases by the power of 2 (y = n^2) but for Plan 2, the number increases by the power of 3 (y = n^3). C. Write an equation for the relationship between the number of the square, n, and the number of rubas, r for plan 2. equation -> r = n^x D. Is the total reward under the king's plan ggreater than or less than the total reward under the peasant's plan? How did you decide? The total reward under the King's plan is less than the total reward under the peasant's plan because although the king triples everytime, he only ends up on the 16th square. And even though the peasant doubles everytime, she ends up on the 64th square. This means she is losing a whole chunk of rubas.
 * Problem 1.3**
 *  **__Square__**  || **__Plan 1 __** || **__Plan 2 __** ||
 * 1 || 1  || 1  ||
 * 2 || 2  || 3  ||
 * 3 || 4  || 9  ||
 * 4 || 8  || 27  ||
 * 5 || <span style="font-family: 바탕;">16  || <span style="font-family: 바탕;">81  ||
 * <span style="font-family: 바탕;">6 || <span style="font-family: 바탕;">32  || <span style="font-family: 바탕;">243  ||
 * <span style="font-family: 바탕;">7 || <span style="font-family: 바탕;">64  || <span style="font-family: 바탕;">729  ||
 * <span style="font-family: 바탕;">8 || <span style="font-family: 바탕;">128  || <span style="font-family: 바탕;">2187  ||
 * <span style="font-family: 바탕;">9 || <span style="font-family: 바탕;">256  || <span style="font-family: 바탕;">6561  ||
 * <span style="font-family: 바탕;">10 || <span style="font-family: 바탕;">512  || <span style="font-family: 바탕;">13122  ||
 * <span style="font-family: 바탕;">11 || <span style="font-family: 바탕;">1024  || <span style="font-family: 바탕;">39366  ||
 * <span style="font-family: 바탕;">12 || <span style="font-family: 바탕;">2048  || <span style="font-family: 바탕;">118098  ||
 * <span style="font-family: 바탕;">13 || <span style="font-family: 바탕;">4036  || <span style="font-family: 바탕;">354294  ||
 * <span style="font-family: 바탕;">14 || <span style="font-family: 바탕;">8192  || <span style="font-family: 바탕;">1062882  ||
 * <span style="font-family: 바탕;">15 || <span style="font-family: 바탕;">16384  || <span style="font-family: 바탕;">3188646  ||
 * <span style="font-family: 바탕;">16 || <span style="font-family: 바탕;">32768  || <span style="font-family: 바탕;">9565938  ||

1.Make a graph of plan 2 for n = 1 to 10. How does your graph compare to the graph you made for plan 1?
 * Follow- up**
 * <span style="font-family: 바탕;"> **__Square__**  || <span style="font-family: 바탕;">  || **__<span style="font-family: 바탕;">Plan 2 __** ||
 * <span style="font-family: 바탕;">1 || <span style="font-family: 바탕;">  || <span style="font-family: 바탕;">1  ||
 * <span style="font-family: 바탕;">2 || <span style="font-family: 바탕;">  || <span style="font-family: 바탕;">3  ||
 * <span style="font-family: 바탕;">3 || <span style="font-family: 바탕;"> || <span style="font-family: 바탕;">9  ||
 * <span style="font-family: 바탕;">4 || <span style="font-family: 바탕;">  || <span style="font-family: 바탕;">27  ||
 * <span style="font-family: 바탕;">5 || <span style="font-family: 바탕;"> || <span style="font-family: 바탕;">81  ||
 * <span style="font-family: 바탕;">6 || <span style="font-family: 바탕;"> || <span style="font-family: 바탕;">243  ||
 * <span style="font-family: 바탕;">7 || <span style="font-family: 바탕;"> || <span style="font-family: 바탕;">729  ||
 * <span style="font-family: 바탕;">8 || <span style="font-family: 바탕;">  || <span style="font-family: 바탕;">2187  ||
 * <span style="font-family: 바탕;">9 || <span style="font-family: 바탕;"> || <span style="font-family: 바탕;">6561  ||
 * <span style="font-family: 바탕;">10 || <span style="font-family: 바탕;"> || <span style="font-family: 바탕;">13122  ||

If they ended up in 10, plan 1 would be way way smaller than plan b. Plan 1 would have 512 rubas and Plan 2 would have 13122 rubas. Plan 2 would have about 25.63 times more rubas than plan 1.

//The queen devised a third reward that the king would offer the peasant. Under plan 3, the king would make a board with 12 squaress. He would start with 1 ruba in the first square and quadruple the number of rubas from one square to the next. So. the pattern of rubas would be 1, 4, 16, 64 and so on.//

2. Make a graph of plan 3 for n = 1 to 10. How does this graph compare to the graphs of plans 1 and 2? <span style="font-family: 바탕;">Square || <span style="font-family: 바탕;">Plan 3 || <span style="font-family: 바탕;">1 || <span style="font-family: 바탕;">1 || <span style="font-family: 바탕;">2 || <span style="font-family: 바탕;">4 || <span style="font-family: 바탕;">3 || <span style="font-family: 바탕;">16 || <span style="font-family: 바탕;">4 || <span style="font-family: 바탕;">64 || <span style="font-family: 바탕;">5 || <span style="font-family: 바탕;">256 || <span style="font-family: 바탕;">6 || <span style="font-family: 바탕;">1024 || <span style="font-family: 바탕;">7 || <span style="font-family: 바탕;">4096 || <span style="font-family: 바탕;">8 || <span style="font-family: 바탕;">16384 || <span style="font-family: 바탕;">9 || <span style="font-family: 바탕;">65536 || <span style="font-family: 바탕;">10 || <span style="font-family: 바탕;">262144 ||

They all are fomed in the same equation r = n^x except that plan 1 is x=2, plan 2 x=3, and plan 3 is x=4.

3. Write an equation for the relationship between the number of the square, n, and the number of rubas, r, for plan 3. How do the equations fot the three plans compare?

4. Of the three plans, which is the best fot the peasant? Which is the best for the King?

5. Design another reward plan with a pattern of change that you think shows exponential growth. Explain why you think the growth is exponential.