Question

In the United States, there were approximately 2.23 million marriages in 2005, compared to 2.28 million in 2004. Use an exponential function to predict the number of marriages in 2025, and discuss the reasonableness of the result.

Answer 1

An exponential function f(x) is in the form of \(ab^x\)
Let \(y=f(t)\) denote the marriages in millions over the years from 2004, \(t\).
Let us consider \(f(0)\) as the number of marriages in 2004:
\(f(0)=ab^0=2.28\)
\(∴a=2.28\)
Now let us find the value of \(b\), by plugging in \(f(1)\):
\(f(1)=ab^1=2.23\)
we determined that a=2.28
\(2.28b=2.23\)
\(b=\_\_\_\)

Determine the value of \(b\) and join the puzzle pieces together to have the correct form of the exponential function. For discussing the reasonableness of the result, you can observe that the number of marriages have decreased from 2004 to 2005, and you will find that \(b<1\) indicating a decay function, which is reasonable.

Answer 2

Once you find the value of \(b\), you must also predict the number of marriages in 2025. We determined that \(t\) is the years from 2004, hence in the year of 2025, \(t=25\). Then just find \(f(25)\), and you should highlight that it is reasonable because it would be a decrease, which follows the original trend.