The number N of farms in the United States has declined...

The number $N$ of farms in the United States has declined continually since 1950 . In $1950,$ there were 5.6 million farms, and in 2008 that number had decreased to 2.2 million Assuming that the number of farms decreased according to the exponential decay model: a) Find the value of $k$, and write an exponential function that describes the number of farms, in millions, after time $t$, in years, where $t$ is the number of years since 1950 . b) Estimate the number of farms in 2011 and in 2015 c) At this decay rate, when will only $1,000,000$ farms remain?

The number $N$ of farms in the United States has declined continually since 1950 . In $1950,$ there were 5.6 million farms, and in 2008 that number had decreased to 2.2 million Assuming that the number of farms decreased according to the exponential decay model:
a) Find the value of $k$, and write an exponential function that describes the number of farms, in millions, after time $t$, in years, where $t$ is the number of years since 1950 .
b) Estimate the number of farms in 2011 and in 2015 c) At this decay rate, when will only $1,000,000$ farms remain?

Key Concepts

Use of Logarithms in Exponential Models

Logarithms are essential in solving exponential decay problems because they allow for the exponent containing the unknown variable to be isolated. By taking the natural logarithm of both sides of the equation, one can linearize the exponential relationship, making it possible to solve for variables such as the decay constant or time.

Extrapolation and Prediction Using Models

Once an exponential decay function has been determined, it can be used to predict or estimate future or past values of the quantity. This model extrapolates data by substituting different time values into the function, providing estimates based on the assumption that the decay process continues at a constant percentage rate.

Decay Constant (k)

The decay constant, denoted by k, quantifies the rate at which the quantity decreases over time. It is derived from known data points by rearranging the exponential decay formula and typically calculated using logarithms. A larger k indicates a faster rate of decay.

Exponential Decay Model

An exponential decay model describes a process where a quantity decreases at a rate proportional to its current value. This is typically mathematically represented by a function of the form N(t) = N? * e^(–kt), where N? is the initial value, k is a positive constant that represents the rate of decay, and t is time.

Transcript

00:01 All right, we're given a situation where the number of farms in the united states has declined continually since 1950.

00:06 So in 1950, there were 5 .6 million farms, so we're going to use that as our initial value.

00:11 And then in 2008, that number had decreased to 2 .2 million.

00:15 So we're going to use that as our output that we're comparing that to.

00:19 And assuming the number of farms decreased according to an exponential decay model, so we're going to be writing a model using p sub -0 or p -not equals negative e.

00:30 Time e to the negative k t all right so in this situation our initial value is going to be 5 .6 million i'm just going to use a 5 .6 5 .6 million e to the negative k k is not given in fact part a is fine the k value and run an exponential function that describes the number of farms in millions in time after 10.

01:01 Time t in years where t is the number of years since 1950.

01:04 So 1950 to 2008 is going to be 58 years.

01:11 So in this case, we do have this value of 58.

01:14 And then our output, after 58 years, it has decreased to 2 .2 million here.

01:22 So we have to figure out what k is.

01:27 And we're going to do that by solving with logarithms.

01:30 So let's start by taking 2 .2, dividing both sides by 5 .6 to isolate e.

01:36 I'm going to use 2 .2.

01:40 Let's just round that to four decimal places.

01:43 And we're going to have 0 .392929 equals e to the negative k times 58.

01:54 So let's take the natural logarithm of both sides.

01:59 When we do that e cancels and we have negative k times 58 we can get that negative k by itself or actually k by dividing by negative 58 so what we have here is the natural log of 0 .3929 over negative 58 so that should tell us what k is let's see what that comes out to be the natural log of 0 .392929 divided by negative 58 is 0 .0161.

02:32 So if k is 0 .0161, that is 1 .61 percent decay rate.

02:43 So let's write out this formula.

02:46 P of t is going to equal 5 .6e to the negative 0 .0161t.

02:55 So there is our model that we can use.

02:59 That to estimate the number of farms in 2011 and 2015.

03:03 Keep in mind that our t value is since 1950...

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