Population Growth of Rabbits. Under ideal econditions, a...

Population Growth of Rabbits. Under ideal econditions, a population of rabbits has an exponenial growth rate of $11.7 \%$ per day. Consider an initial population of 100 rabbits. a) Find the exponential growth function. (3) Graph the function. c) What will the population be after 7 days? after 2 weeks? Wind the doubling time.

Population Growth of Rabbits. Under ideal econditions, a population of rabbits has an exponenial growth rate of $11.7 \%$ per day. Consider an initial population of 100 rabbits.
a) Find the exponential growth function.
(3) Graph the function.
c) What will the population be after 7 days? after
2 weeks?
Wind the doubling time.

Key Concepts

Doubling Time

Doubling Time is the amount of time required for a quantity undergoing exponential growth to double in size. It can be calculated using the growth rate, and it provides a useful measure of how quickly the exponential process is accelerating.

Graphing Exponential Functions

Graphing Exponential Functions involves plotting the function on a coordinate system to visualize how the quantity changes over time. These graphs typically show a slow initial increase followed by rapid growth, reflecting the multiplicative nature of exponential change.

Exponential Growth Function

An Exponential Growth Function is a mathematical representation of exponential growth, typically expressed as P(t) = P? * e^(rt) or P(t) = P? * (1 + r)^t. Here, P? is the initial value, r is the constant growth rate per time unit, and t represents time.

Exponential Growth

Exponential Growth describes a process where a quantity increases by a fixed percentage or factor over equal time intervals. This means that the rate of growth is proportional to the current amount, leading to rapid increases after sufficient time.

Growth Rate

The Growth Rate represents the constant percentage increase in each time period in an exponential growth scenario. It is a crucial parameter in determining how quickly the quantity increases and is often expressed in decimal form when used in exponential functions.

Transcript

00:01 In this problem, we're given a population of rabbits with the initial population b0 of 100, and the growth rate is exponential with a rate of 11 .7 % per day.

00:20 So let's convert the percentage into decimal, and we can do that by dividing by 100, and we get 0 .117 as r.

00:32 Now the standard form for an exponentially growing population is given by the formula b of t, which is population at any given time equals p .0 times e to the rt.

00:50 Now p0 is just 100, so the initial population, and it's raised to the power e of rt, and r is 0 .117.

01:02 So our population function as a function of time equals, p of t equals 100 times e to the 0 .117t.

01:18 Now for the second part, we need to graph this function.

01:27 So we need to make a rough sketch of the function.

01:41 So we can have our horizontal axis of t and our vertical axis as p of t.

01:54 So it starts at 100.

02:03 So from here we can calculate a few values.

02:07 And a few useful ones will be the multiple times.

02:13 Of 5, so pf 0 is just the initial population, which is 100.

02:20 So we'll plot that point.

02:24 P of 5 is equal to 100 times e to the 0 .117 times 5, which equals 179.

02:36 We can round that off as 180, and we can plot that point somewhere over here.