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A growing farming conglomerate increases its water usage at a rate of $10\%$ every year. If it used $13,000$ megaliters of water this year, then how much water will it use $17$ years from now? If necessary, round your answer to the nearest hundredth. $\newline$ $\_\_\_\_$ megaliters

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Exponential growth and decay: word problems

Full solution

Q. A growing farming conglomerate increases its water usage at a rate of

10\%

every year. If it used

13,000

megaliters of water this year, then how much water will it use

17

years from now? If necessary, round your answer to the nearest hundredth.

\newline

\_\_\_\_

megaliters

Identify initial amount and rate: Identify the initial amount of water used and the rate of increase. $\newline$ The initial amount of water used is $13,000$ megaliters, and the rate of increase is $10\%$ per year.
Convert percentage to decimal: Convert the percentage increase to a decimal to use in calculations. $\newline$ A $10\%$ increase is the same as multiplying by $1.10$ (since $10\%$ as a decimal is $0.10$ , and you add $1$ to account for the original amount).
Determine number of years: Determine the number of years over which the water usage will increase. $\newline$ The number of years is given as $17$ .
Use exponential growth formula: Use the formula for exponential growth to calculate the future water usage. $\newline$ The formula is $P(t) = P_0 \times (1 + r)^t$ , where $P(t)$ is the future amount, $P_0$ is the initial amount, $r$ is the rate of increase, and $t$ is the time in years.
Substitute values and calculate: Substitute the known values into the formula and calculate the future water usage. $\newline$ $P(17) = 13,000 \times (1.10)^{17}$
Final calculation: Calculate the future water usage using the values from Step $5$ . $\newline$ $P(17) = 13,000 \times (1.10)^{17}$ $\newline$ $P(17) = 13,000 \times 4.801$ (rounded to three decimal places) $\newline$ $P(17) \approx 62,413$ megaliters (rounded to the nearest hundredth)

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