Modeling is a method of simulating the real life situations with mathematical equations. Using these models we can forecast the future behavior. We can translate the mathematical word problem into a math expression using variables.

Calculation:

The given model is \(f{{\left({x}\right)}}={13},{017}{\left({1.05}\right)}^{x}\)

Where x is the number of years since 2006 and y = f(x) is the cost in dollars.

From the table, the average annual cost in 2006 is $12,837.

After xyears the 2006 will be doubled

So, After xyears, the cost will be \({2}\times{12837}=\${25674}\).

Hence, \(f{{\left({x}\right)}}={13},{017}{\left({1.05}\right)}^{x}={25674}\)

\({\left({1.05}\right)}^{x}=\frac{25674}{{13017}}\)

Taking natural logarithm on each side

\(\ln{\left({1.05}\right)}^{x}=\ln\frac{25674}{{13017}}\)

Using calculator, \({x}{\left({I}{n}{1.05}\right)}={0.679}\)

Divide by \(\ln{{1.05}},{x}=\frac{0.679}{ \ln{{1.05}}}\)

Using calculator, x = 13.92

Rounded off to nearest tens, \(x \approx 13\)

Hence, based on this model the 2006 cost will be doubled in 13 years since 2006.

That is, the year = 2006+13 = 2019

Final statement:

Hence, based on this model the 2006 cost will be doubled in 2019.