Question 15
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The population of bats in a large cave is 6600 and is growing exponentially at $6 \%$ per year. Write a function to represent the population of bats after $t$ years, where the monthly rate of change can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage rate of change per month, to the nearest hundredth of a percent.
Answer Attempt 1 out of 2
Function: $f(t)=$
Growth
$\%$ increase per month
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Step 1 ：The problem is asking for a function that represents the population of bats after t years, given that the population is growing exponentially at 6% per year. We also need to find the monthly rate of change.

Step 2 ：To solve this, we can use the formula for exponential growth, which is \(P(t) = P0 * e^{rt}\), where \(P0\) is the initial population, \(r\) is the rate of growth, and \(t\) is time.

Step 3 ：In this case, \(P0\) is 6600, \(r\) is 6% or 0.06, and \(t\) is the number of years.

Step 4 ：To find the monthly rate of change, we can divide the annual rate of change by 12.

Step 5 ：The function to represent the population of bats after t years is \(f(t) = 6600 * e^{0.06t}\). The monthly rate of change is 0.5%.

Step 6 ：Final Answer: The function is \(f(t) = 6600 * e^{0.06t}\) and the monthly rate of change is \(\boxed{0.5\%}\).