ossidianaZ

2021-08-11

To find: The year in which the 2006 cost of tution, room and board fees in public colleges will be doubled using the function $f\left(x\right)=13.017{\left(1.05\right)}^{x}$

BleabyinfibiaG

Step 1
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 $=f\left(x\right)$ is the cost in dollars.
From the table, the average annual cost in 2006 is $\mathrm{}12.837.$
After x years the 2006 will be doubled
So, After x years, 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, $x\left(\mathrm{ln}1.05\right)=0.679$
Divide by
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 doublend in 2019.

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