Math / Calculus

QuestionA bacteria cylture starts with 220 bacteria and grows at a rate proportional to its size. After 6 hours the population doubled. (a) Express the population $A$ after $t$ hours as a function of $t$ . (b) What will be the population after 7 hours? (c) How long will it take for the population to reach 2530? (a) Express the population $A$ after $t$ hours as a function of $t$ . $A(t)=$ $\square$ Round $k$ to 4 decimal places.) (b) What will be the population after 7 hours?
Approximately $\square$ bacteria. (Do not round until the final answer. Then round to the nearest whole number as needed.) (c) How long will it take for the population to reach 2530? in approximately $\square$ hours. (Do not round until the final answer. Then round to the nearest hundredth as needed.)

Studdy Solution
To find the time when the population reaches 2530, set $A(t) = 2530$ :
$2530 = 220 e^{0.1155t}$
Solve for $t$ :
$\frac{2530}{220} = e^{0.1155t}$
$11.5 = e^{0.1155t}$
$\ln(11.5) = 0.1155t$
$t = \frac{\ln(11.5)}{0.1155}$
Calculate $t$ :
$t \approx \frac{2.4423}{0.1155}$
$t \approx 21.14$
Approximately 21.14 hours.
The solutions are: (a) $A(t) = 220 e^{0.1155t}$ (b) Approximately 494 bacteria after 7 hours. (c) Approximately 21.14 hours to reach a population of 2530.

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