Exponential and logarithmic equation

Question

Please help I don't understand what to do. 

 

In 1990, there were 33,500 farms in North Dakota. In 2000, there were 30,800.
A.) Find the value of k for the exponential function N(t)=N0e^kt to model the number of farms.
B.) Use your model to predict the number of farms in North Dakota in 2010.

Answer 1

Hello, thank you for taking the time to post your question!

The underlying equation here is

N(t) = N0 * e^(kt)

Plugging in the values we know that becomes

30800 = 33500 * e^(10k)

If you solve that algebraically for the value of “k” using logarithms it becomes

30800/33500 = e^(10k)

k = ln(30800/33500)/10

k = -0.0084

So now we have the general from of the equation

N(t) = 33500*e^(-0.0084t)

For the second part you want to plug in t = 2010 – 1990 = 20 and then evaluate! When I do that I get 28,318 for the predicted value.

I hope that helps you get moving in a better direction on this type of question! Feel free to reach out if you have any additional questions beyond that :)

Answer 2

In 1990, there were 33,500 farms in North Dakota. In 2000, there were 30,800.

A.) Find the value of k for the exponential function N(t)=N(0)e^(kt) to model the number of farms.

Let t be the number of years since 1990.

33500 = N(0)e^(k*0) = N(0)

N(2000-1990) = 30800 = 33500e^(k(10))

e^(k(10)) = 308/335

k = (ln(308/335))/10 ≈ -0.00840307488515


B.) Use your model to predict the number of farms in North Dakota in 2010.

N(2010-1990) = N(20) = 33500 e^(-0.00840307488515*20)

N(20) ≈ 28317.6119402980535

There must be an integer number of farms; but do you round up as usual, or down because the decimals don’t represent a full farm?