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Suatu bakteri dapat membelah diri menjadi tiga setiap $13$ menit. Jika banyak bakteri mula-mula berjumlah $20$ , diperlukan waktu $t$ agar jumlah bakteri menjadi $14.580$ . Jika bakteri tersebut membelah diri menjadi tiga setiap $26$ menit, banyak bakteri setelah waktu $t$ adalah ....

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Find the sum of a finite geometric series

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Q. Suatu bakteri dapat membelah diri menjadi tiga setiap

13

menit. Jika banyak bakteri mula-mula berjumlah

20

, diperlukan waktu

t

agar jumlah bakteri menjadi

14.580

. Jika bakteri tersebut membelah diri menjadi tiga setiap

26

menit, banyak bakteri setelah waktu

t

adalah ....

Identify initial and final amounts: Identify the initial amount of bacteria $a_1$ and the final amount after time $t$ when tripling every $13$ minutes $a_t$ . $a_1 = 20$ , $a_t = 14,580$
Determine number of triplings: Determine the number of triplings $n$ that occur in time $t$ when the bacteria triple every $13$ minutes. $\newline$ Use the formula for geometric growth: $a_t = a_1 \times r^{n}$ , where $r$ is the common ratio ( $3$ in this case). $\newline$ $14,580 = 20 \times 3^n$
Divide to isolate $3^n$ : Divide both sides by $20$ to isolate $3^n$ . $\newline$ $3^n = \frac{14,580}{20}$ $\newline$ $3^n = 729$
Find value of $n$ : Find the value of $n$ by recognizing that $729$ is a power of $3$ . $3^6 = 729$ $n = 6$
Calculate triplings for $26$ minutes: Now, calculate the number of triplings $n_{26}$ that occur in time $t$ when the bacteria triple every $26$ minutes. $\newline$ Since the tripling period is doubled from $13$ to $26$ minutes, the number of triplings in the same time $t$ will be halved. $\newline$ $n_{26} = n / 2$ $\newline$ $n_{26} = 6 / 2$ $\newline$ $n_{26} = 3$
Calculate amount of bacteria: Calculate the amount of bacteria $a_{26}$ after time $t$ when tripling every $26$ minutes. $\newline$ Use the formula $a_{26} = a_1 \cdot r^{n_{26}}$ . $\newline$ $a_{26} = 20 \cdot 3^3$ $\newline$ $a_{26} = 20 \cdot 27$ $\newline$ $a_{26} = 540$

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