Question

# To find:The year in which the 2006 cost of tuition, room and board fees in public colleges will be doubled using the function f{{left({x}right)}}={13},{017}{left({1.05}right)}^{x}.

Math Word Problem
To find:The year in which the 2006 cost of tuition, room and board fees in public colleges will be doubled using the function $$f{{\left({x}\right)}}={13},{017}{\left({1.05}\right)}^{x}$$.

2021-02-04
Concept:
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.