Question A pride of lions is growing at a rate of $0.5 \%$ per year, compounded continuously. If the growth rate continues, how many years will it take for the size of the pride to reach $275 \%$ of its current size? Round your answer up to the nearest whole number, and do not include units. Provide your answer below:
Solution
Step 1 ：We are given that a pride of lions is growing at a rate of 0.5% per year, compounded continuously. We are asked to find out how many years it will take for the size of the pride to reach 275% of its current size.
Step 2 ：We can use the formula for continuous compounding, which is \(A = Pe^{rt}\), where \(A\) is the final amount, \(P\) is the principal amount, \(r\) is the rate of growth, and \(t\) is the time.
Step 3 ：In this case, we know \(A\) (275% of the current size), \(P\) (the current size, which we can consider as 1 for simplicity), and \(r\) (0.5% per year). We need to solve for \(t\).
Step 4 ：Substituting the given values into the formula, we get \(2.75 = 1e^{0.005t}\).
Step 5 ：Solving for \(t\), we get \(t = 203.0\).
Step 6 ：Rounding up to the nearest whole number, we get \(t = 203\).
Step 7 ：Final Answer: It will take \(\boxed{203}\) years for the size of the pride to reach 275% of its current size.
