Deer Population The graph shows the deer population in a...

Deer Population The graph shows the deer population in a Pennsylvania county between 2010 and $2014 .$ Assume that the population grows exponentially. $$ \begin{array}{l}{\text { (a) What was the deer population in } 2010 ?} \\ {\text { (b) Find a function that models the deer population } t \text { years }} \\ {\text { after } 2010 \text { . }} \\ {\text { (c) What is the projected deer population in } 2018 ?}\end{array} $$$$ \begin{array}{l}{\text { (d) Estimate how long it takes the population to reach }} \\ {100,000 .}\end{array} $$

Deer Population The graph shows the deer population in a
Pennsylvania county between 2010 and $2014 .$ Assume that
the population grows exponentially.

$$
\begin{array}{l}{\text { (a) What was the deer population in } 2010 ?} \\ {\text { (b) Find a function that models the deer population } t \text { years }} \\ {\text { after } 2010 \text { . }} \\ {\text { (c) What is the projected deer population in } 2018 ?}\end{array}
$$$$
\begin{array}{l}{\text { (d) Estimate how long it takes the population to reach }} \\ {100,000 .}\end{array}
$$

Key Concepts

Solving Exponential Equations

Solving exponential equations often requires the use of logarithms to isolate variables, particularly when determining the time required to reach a specific value. This technique is fundamental when finding how long it will take for an exponentially growing process to achieve a predetermined level, making it a vital tool in various application areas such as population studies and finance.

Exponential Growth

Exponential growth describes a process in which a quantity increases at a rate proportional to its current value. This concept is commonly used to model populations, investments, or any system where the change over time is multiplicative rather than additive, meaning the larger the population, the faster it grows.

Exponential Function Modeling

An exponential function is typically written in the form P(t) = P? · a? or P(t) = P? · e^(kt), where P? represents the initial value, a or e^(k) is the growth factor, and t is time. This function is used to model situations where growth or decay occurs continuously over time, reflecting the inherent multiplicative nature of the process.

Initial Value Identification

Identifying the initial value, often denoted as P?, is crucial in setting up any exponential model. It represents the quantity at the starting point (t=0) and serves as a baseline from which all future growth is calculated. This value is typically obtained directly from given data or observation.

Parameter Estimation

Parameter estimation involves determining the values of the unknown constants in the exponential model, such as the growth rate (k) or the base (a). This is usually achieved by using known data points from the observed process, which can then be used to calibrate the model to fit real-world data.

Extrapolation and Prediction

Extrapolation uses the established exponential model to predict future values beyond the range of the observed data. This process involves substituting a future time value into the model function to estimate the quantity at that time, which is essential in planning and resource management scenarios.

Transcript

00:01 So this from, uh is modeling a deer population and just reading it off the graph.

00:05 I can see that on.

00:06 And for part a just asked, what's the deer population? 2010.

00:10 I'm reading that off the graph.

00:11 It's just we want t value of zero.

00:14 And so the team volume zero gives us 20,000 dear.

00:22 Okay, part b says find a function.

00:26 That model's the deer population t years after 2010.

00:33 Okay, so to find on exponential function for this, we know it's gonna be in the form.

00:39 Uh, end of t equals our, um i guess are in not is gonna be 20,000 and then times e to the r see.

00:56 So we need to find my heart is so do that.

00:59 I'm gonna plug in my other point.

01:02 We have another point at 31,000.

01:04 So we have 31,000 equals 20,000 times e to the r.

01:16 And then the t value was four.

01:25 Okay, if i divide both sides by 20,000 let's see, i get 31,000 maybe around 1.5.

01:44 We'll have 1.55 because deed to the four tongues are i don't see the natural long about science.

01:55 So have the natural lago.

01:57 1.55 eagles.

02:01 Four times are and then just divide besides by four i don't know are is equal to the natural log of 1.55 all over for you.

02:16 You just put that in a calculator that's gonna give me 0.1096 uh, okay, so, for part b, i'm a final equation can be n f t because 20,000 times e to the 0.1096 times t is this part b part? see, i could start any sign says find a fun.

03:21 What is the projected deer population? 2018...

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