Question

The average population starts at 600 rabbits and increases by 5% each month. Find an equation for the population, P, in terms of the months since January.

Answer 1

The equation for the population, P, in terms of the months since January is P = 600(1 + 0.05)^t.

Step-by-step explanation:

The equation for the population, P, in terms of the months since January can be found using the formula for compound interest:

P = P0(1 + r)t

Where P0 is the initial population (600 rabbits in this case), r is the monthly growth rate (5% or 0.05), and t is the number of months since January.

Therefore, the equation for the population is P = 600(1 + 0.05)t.

Answer 2

The equation for the population, P, in terms of the months since January, denoted by
`\(t\)`, starting at 600 rabbits and increasing by 5% each month is:

`\[ P(t) = 600 * (1 + 0.05)^t \]`

Step-by-step explanation:

To find an equation for the population growth over time, the initial population of 600 rabbits, increasing by 5% each month, is considered. The general formula for exponential growth is
`\( P(t) = P_0 * (1 + r)^t \)`, where:

-
`\( P(t) \)` represents the population at time 't'.

-
`\( P_0 \)` is the initial population.

-
`'r'` is the growth rate.

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`'t'` is the time in months.

In this scenario, the initial population is 600 rabbits, and the growth rate per month is 5%. Plugging these values into the formula gives:

`\[ P(t) = 600 * (1 + 0.05)^t \]`

`\[ P(t) = 600 * 1.05^t \]`

This equation illustrates how the population, P, changes over time, represented by 't' months since January. Each month, the population increases by 5% of the previous month's population, leading to an exponential growth pattern.