Number of Farms. The number N of farms in the United States...

Number of Farms. 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 (Sources: U.S. Department of Agriculture; National Agricultural Statistics Service). 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 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?

Number of Farms. 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 (Sources: U.S. Department of Agriculture; National Agricultural Statistics Service).
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 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

Exponential Decay

Exponential decay is a process in which the value of a quantity decreases at a rate proportional to its current value. This results in the quantity reducing by a constant percentage over equal time intervals, and is mathematically modeled using an exponential function with a negative exponent.

Exponential Function Model

An exponential function model typically has the form N(t) = N?e^(kt) where N? is the initial value, k is a constant, and t represents time. When modeling decay, the constant k is negative, or alternatively the model can be written to show decay explicitly, such as N(t) = N?e^(–kt).

Parameter Estimation in Exponential Models

Parameter estimation in exponential models involves determining the constants such as N? and k using known data points. Given values at specific times, one sets up equations to solve for these parameters, ensuring the model accurately represents the observed behavior of the quantity.

Prediction Using Exponential Models

After determining the parameters of an exponential decay model, the function can be used both to predict future values of the quantity at specified times and to interpolate values between data points. This technique is widely applied in various fields including physics, biology, and economics.

Solving for Time in Exponential Equations

To find the time at which the quantity reaches a specific value, one must solve the exponential equation for time. This typically involves taking the natural logarithm of both sides of the equation to isolate the time variable, and then solving the resulting linear equation.

Transcript

00:01 So we have a question over here given that the population all this represent population population count in millions so 1950 the population was 5 .6 million and 2008 the population was 2 .2 million so it follows radioactive decay so i'll say that p of t is equal to p p .0 e -raise to the power minus k t so using the given information i'll say that 2 .2 million is equal to 5 .6 million into e -res to the power minus k into the time is 58 years so let's keep it like that so the first part of the question they are asking the rate of the decay so i'm going to write it as minus 58k the exponential value i'm writing it directly as ln of this is 2 .2 divided by 5 .6 basically numbers these are just million million will get cancelled so from here when i do the calculations i'll get this ln value is equal to minus 0 .934 and when i divide by 58 i'll get the k value is equal to 0 .01 61 so if i convert this rate this rate is actually per year all right so when i convert it into percentage implies k is equal to 1 .61 percentage per year so that's the rate of decay that i'm having now this is the first part actually if i go to the second part in the second part they're asking the population in the year 2011 so for that i'm going to use this formula directly directly i can put this formula so p -0 is 5 .6 million into e -res to the power minus k t so i can say minus k t so minus k value is i'll use this k value so that is minus 0 .0161 multiplied by here i'll multiply it with 51 so when i do the multiplication i'll have this value as so on multiplying and putting all this values i'm going to get this value to be 2 .4637 millions and when somebody is asking me the same population in the year 2015 so here i am supposed to replace that time period with 55 years that is minus 0 .0161 multiplied by 55 so on multiplying this value i am going to get 2 .31 million so that's the answer for the b part there is this tiny mistake instead of writing here 51 it should have been 61 years actually so with 61 years i am going to have a population of 2 .097 million.

03:24 And here instead of 61 i'm going to write 65 years.

03:27 So this population is supposed to be 1 .966 millions...

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