2021-02-03

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

odgovoreh

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 $2×12837=\mathrm{}25674$.
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
$\mathrm{ln}{\left(1.05\right)}^{x}=\mathrm{ln}\frac{25674}{13017}$
Using calculator, $x\left(In1.05\right)=0.679$
Divide by $\mathrm{ln}1.05,x=\frac{0.679}{\mathrm{ln}1.05}$
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.

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