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A nautilus shell is made up of many chambers, each chamber roughly $5\%$ larger than the previous one. Assuming a nautilus creates a new chamber every year, and this year's chamber has a volume of $880$ microliters, how large will the chamber created in $8$ years be? If necessary, round your answer to the nearest tenth. $\newline$ $\newline$ ____ microliters $\newline$

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Exponential growth and decay: word problems

Full solution

Q. A nautilus shell is made up of many chambers, each chamber roughly

5\%

larger than the previous one. Assuming a nautilus creates a new chamber every year, and this year's chamber has a volume of

880

microliters, how large will the chamber created in

8

years be? If necessary, round your answer to the nearest tenth.

\newline

\newline

____ microliters

\newline

Identify initial volume and growth rate: Identify the initial volume and the rate of growth. $\newline$ The initial volume of the chamber is $880$ microliters, and the rate of growth is $5\%$ larger than the previous one each year.
Convert percentage growth to decimal: Convert the percentage growth to a decimal. $\newline$ To work with the percentage in calculations, convert it to a decimal. A $5\%$ increase is the same as multiplying by $1.05$ (since $5\%$ as a decimal is $0.05$ , and you add this to $1$ to account for the original volume).
Determine formula for volume: Determine the formula for the volume after a certain number of years. $\newline$ The formula for the volume after $n$ years, given a constant percentage increase, is $V(n) = V(0) \times (1 + r)^n$ , where $V(0)$ is the initial volume, $r$ is the rate of growth as a decimal, and $n$ is the number of years.
Substitute values into formula: Substitute the known values into the formula. $V(8) = 880 \times (1.05)^8$
Calculate volume after $8$ years: Calculate the volume after $8$ years. $\newline$ $V(8) = 880 \times (1.05)^8$ $\newline$ Using a calculator, we find: $\newline$ $V(8) \approx 880 \times 1.477455$ $\newline$ $V(8) \approx 1300.1604$ microliters
Round answer to nearest tenth: Round the answer to the nearest tenth. $\newline$ The volume of the chamber created in $8$ years, rounded to the nearest tenth, is approximately $1300.2$ microliters.

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