a person was modeling a growing population with a growth factor of 1.5 using pennies. they started with 4 pennies and flipped all 4 pennies. for every penny that landed heads up another penny was added to the pile. then, the new pile of pennies are all flipped and again for each one landing on heads another penny was added. the process is repeated several more times.
which of the below would be the most reasonable number of pennies to be expected to be in the pile after pennies were added just after the 5th flip?
a. 6 pennies
b. 14 pennies
c. 30 pennies
d. 100 pennies
4*(3/2)^5
30
To determine the number of pennies in the pile after the 5th flip, we need to calculate the growth factor at each stage:
1st flip: Starting with 4 pennies.
2nd flip: Each penny landed heads, so 4 more pennies are added (4 + 4 = 8).
3rd flip: Each of the 8 pennies landed heads, so 8 more pennies are added (8 + 8 = 16).
4th flip: Each of the 16 pennies landed heads, so 16 more pennies are added (16 + 16 = 32).
5th flip: Each of the 32 pennies landed heads, so 32 more pennies are added (32 + 32 = 64).
Therefore, the most reasonable number of pennies to be expected in the pile after the 5th flip is 64 pennies.
The correct option is not listed among the options provided.
To solve this problem, we need to understand the concept of exponential growth with a given growth factor. In this case, the growth factor is 1.5, meaning that for each penny that lands heads up, another penny is added to the pile.
Let's break down the problem step by step:
1. Initially, there are 4 pennies in the pile.
2. After the first flip, let's find the number of pennies that land heads up. Since all 4 pennies are flipped, there is a 50% chance for each penny to land heads up. On average, you would expect 50% of the pennies, which is 2 out of 4 pennies, to land heads up.
3. Therefore, after the first flip, there will be 4 + 2 = 6 pennies in the pile.
Now let's further analyze the problem:
4. For each subsequent flip, the number of pennies in the pile doubles, but only for the pennies that landed heads up in the previous flip.
5. So after the second flip, the number of pennies in the pile will be 6 * 1.5 = 9. Since we round to the nearest penny, there are still 9 pennies.
6. After the third flip, the number of pennies in the pile will be 9 * 1.5 = 13.5. Rounding to the nearest penny, there are 14 pennies.
7. After the fourth flip, the number of pennies in the pile will be 14 * 1.5 = 21. Rounding to the nearest penny, there are 21 pennies.
8. After the fifth flip, the number of pennies in the pile will be 21 * 1.5 = 31.5. Rounding to the nearest penny, there are 32 pennies.
Therefore, the most reasonable answer from the given options is d: 100 pennies.