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.

Answer 1

To find the number of years it will take for the size of the pride to reach 275% of its current size, we can use the formula for compound interest. The calculation leads to an approximate value of 122 years.

Step-by-step explanation:

To find the number of years it will take for the size of the pride to reach 275% of its current size, we can use the formula for compound interest. The formula is given by A = P * e^rt, where A is the final size of the pride, P is the initial size of the pride, r is the growth rate expressed as a decimal, and t is the time in years.

In this case, the initial size of the pride is 100% and the growth rate is 0.5% per year, which can be expressed as 0.005. We want to find the value of t when A is 275%. Plugging in the given values, we have 2.75 = 1 * e^(0.005t).

To solve for t, we can take the natural logarithm of both sides of the equation, resulting in ln(2.75) = 0.005t. Dividing both sides by 0.005 gives us t = ln(2.75)/0.005. Using a calculator, we find that t is approximately 122 years.

Answer 2

Step-by-step explanation:

The general formula for continuous exponential growth is A=A0•ert, where A0 is the initial amount, t is the elapsed time, and r is the rate of growth or decay. For this problem, r=0.005, so the formula is A=A0•e0.005t.

Since we want to know how long it takes of the amount to be 275% of the original amount, we can state that A=2.75A0. Substituting that into our formula gives us 2.75A0=A0•e0.005t. Dividing by A0 yields 2.75=e0.005t.

At this point we need to take the natural logarithm of both sides:

ln(2.75)=ln(e0005t)

ln(2.75)=0.005t

t=ln(2.75)/0.005, which is approximately equal to 202.3

We round up to 203 years.