Khadija Wells
2021-02-03
Answered

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

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odgovoreh

Answered 2021-02-04
Author has **107** answers

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$2\times 12837=\mathrm{\$}25674$ .

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.

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

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

Hence,

Taking natural logarithm on each side

Using calculator,

Divide by

Using calculator, x = 13.92

Rounded off to nearest tens,

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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