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The town of Valley View conducted a census this year, which showed that it has a population of $1,800$ people. Based on the census data, it is estimated that the population of Valley View will grow by $12\%$ each decade. $\newline$ Write an exponential equation in the form $y = a(b)^x$ that can model the town population, $y$ , $x$ decades after this census was taken. $\newline$ Use whole numbers, decimals, or simplified fractions for the values of $a$ and $b$ . $\newline$ $y =$ ______

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Write exponential functions: word problems

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Q. The town of Valley View conducted a census this year, which showed that it has a population of

1,800

people. Based on the census data, it is estimated that the population of Valley View will grow by

12\%

each decade.

\newline

Write an exponential equation in the form

y = a(b)^x

that can model the town population,

y

,

x

decades after this census was taken.

\newline

Use whole numbers, decimals, or simplified fractions for the values of

a

and

b

.

\newline

y =

______

Convert to decimal: Convert the percentage growth rate to a decimal. $\newline$ To convert a percentage to a decimal, divide by $100$ . $\newline$ $r = 12\%$ $\newline$ $= \frac{12}{100}$ $\newline$ $= 0.12$
Calculate growth factor: Determine the growth factor $b$ . For exponential growth, the growth factor $b$ is calculated by adding $1$ to the growth rate expressed as a decimal. Growth factor: $b = 1 + r$
Calculate value of growth factor: Calculate the value of the growth factor $b$ . Substitute $r = 0.12$ into the formula $b = 1 + r$ . $b = 1 + 0.12$ $b = 1.12$
Identify initial population and growth factor: Identify the initial population $a$ and the growth factor $b$ to write the exponential equation. $\newline$ The initial population $a$ is given as $1800$ . $\newline$ The growth factor $b$ we calculated is $1.12$ . $\newline$ The exponential equation in the form $y = a(b)^x$ is: $\newline$ $y = 1800(1.12)^x$

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