Bacteria in a culture increase at a rate that is proportional at any time to the number of bacteria at that time. There were 800 bacteria initially, and they triple every 10 hours. What is the number of bacteria after 4 hours? Round to the nearest integer. bacteria

Identify Initial Bacteria Data: Identify the initial number of bacteria and the growth rate. The initial number of bacteria (a) is 800, and they triple every 10 hours.Determine Growth Type: Determine the type of growth. The growth is exponential because the rate of increase is proportional to the number of bacteria present at any time.Calculate Tripled Periods: Calculate the number of times the bacteria will have tripled in 4 hours. Since the bacteria triple every 10 hours, we need to find out how many 10-hour periods are in 4 hours. This is a fraction of a 10-hour period, specifically 4/10 or 0.4 of a period.Use Exponential Growth Formula: Use the exponential growth formula. The formula for exponential growth is P(t) = a * b^(t/T), where P(t) is the population at time t, a is the initial population, b is the growth factor, t is the elapsed time, and T is the time it takes for the population to grow by a factor of b.Substitute Values: Substitute the values into the formula. a = 800 (initial population) b = 3 (growth factor, since the population triples) t = 4 (elapsed time in hours) T = 10 (time it takes to triple) P(4) = 800 * 3^(4/10)Calculate Population: Calculate the population after 4 hours. P(4) = 800 * 3^(0.4) First, calculate 3^(0.4).Evaluate Exponential Value: Evaluate 3^(0.4). Using a calculator, we find that 3^(0.4) is approximately 1.5157.Multiply Initial Population: Multiply the initial population by the growth factor raised to the power calculated in the previous step. P(4) = 800 * 1.5157Perform Multiplication: Perform the multiplication to find the final population. P(4) = 800 * 1.5157 ≈ 1212.56Round Final Result: Round the result to the nearest integer. The number of bacteria after 4 hours, rounded to the nearest integer, is approximately 1213.

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Bacteria in a culture increase at a rate that is proportional at any time to the number of bacteria at that time.
There were 800 bacteria initially, and they triple every 10 hours.
What is the number of bacteria after 4 hours? Round to the nearest integer.
bacteria

Bacteria in a culture increase at a rate that is proportional at any time to the number of bacteria at that time. $\newline$ There were $800$ bacteria initially, and they triple every $10$ hours. $\newline$ What is the number of bacteria after $4$ hours? Round to the nearest integer. $\newline$ $\square$ bacteria

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Q. Bacteria in a culture increase at a rate that is proportional at any time to the number of bacteria at that time.

\newline

There were

800

bacteria initially, and they triple every

10

hours.

\newline

What is the number of bacteria after

4

hours? Round to the nearest integer.

\newline

\square

bacteria

Identify Initial Bacteria Data: Identify the initial number of bacteria and the growth rate. $\newline$ The initial number of bacteria $a$ is $800$ , and they triple every $10$ hours.
Determine Growth Type: Determine the type of growth. The growth is exponential because the rate of increase is proportional to the number of bacteria present at any time.
Calculate Tripled Periods: Calculate the number of times the bacteria will have tripled in $4$ hours. $\newline$ Since the bacteria triple every $10$ hours, we need to find out how many $10$ -hour periods are in $4$ hours. This is a fraction of a $10$ -hour period, specifically rac{4}{10} or $0.4$ of a period.
Use Exponential Growth Formula: Use the exponential growth formula. $\newline$ The formula for exponential growth is $P(t) = a \cdot b^{(t/T)}$ , where $P(t)$ is the population at time $t$ , $a$ is the initial population, $b$ is the growth factor, $t$ is the elapsed time, and $T$ is the time it takes for the population to grow by a factor of $b$ .
Substitute Values: Substitute the values into the formula. $\newline$ $a = 800$ (initial population) $\newline$ $b = 3$ (growth factor, since the population triples) $\newline$ $t = 4$ (elapsed time in hours) $\newline$ $T = 10$ (time it takes to triple) $\newline$ $P(4) = 800 \times 3^{\frac{4}{10}}$
Calculate Population: Calculate the population after $4$ hours. $\newline$ $P(4) = 800 \times 3^{0.4}$ $\newline$ First, calculate $3^{0.4}$ .
Evaluate Exponential Value: Evaluate $3^{0.4}$ . Using a calculator, we find that $3^{0.4}$ is approximately $1.5157$ .
Multiply Initial Population: Multiply the initial population by the growth factor raised to the power calculated in the previous step. $P(4) = 800 \times 1.5157$
Perform Multiplication: Perform the multiplication to find the final population. $\newline$ $P(4) = 800 \times 1.5157 \approx 1212.56$
Round Final Result: Round the result to the nearest integer. $\newline$ The number of bacteria after $4$ hours, rounded to the nearest integer, is approximately $1213$ .