Final answer:
Allyah's earnings on the 15th day of babysitting, where her pay doubles each day from $1, would be $16,384.
Step-by-step explanation:
Allyah is earning money through a process of exponential growth, where her payment doubles each day. To calculate her earnings on day 15, we use the formula for geometric sequences. Since she is paid $1 on the first day (day 0), and the amount doubles each day, the formula to find her earnings on day 15 is:
Earnings on day 15 = Initial amount × (Rate of increase)^(Number of days)
In mathematical terms, this is: $1 × 2^(15-1), since we start counting from day 0. We subtract 1 from the total number of days to adjust for counting the initial payment as day 0.
Earnings on day 15 = $1 × 2^14
Earnings on day 15 = $1 × 16,384
Earnings on day 15 = $16,384
Therefore, Allyah will earn $16,384 on the 15th day of babysitting her cousins.