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Chapter 3: Problem 34

Bacteria Growth The number of bacteria in a culture is increasing according to the law of exponential growth. The initial population is 250 bacteria, and the population after 10 hours is double the population after 1 hour. How many bacteria will there be after 6 hours?

Short Answer

Expert verified
The number of bacteria after 6 hours will be \(250 \times e^{6\times(ln(2)/9)}\)

Step by step solution

01

Set up the equations

Formulate two equations using the given information and the exponential growth formula. Let's denote the growth rate by r. 1) After 1 hour:The population is \(N = 250 \times e^{r}\)2) After 10 hours:The population is \(N = 250 \times e^{10r}\) and it is given to be double the population after 1 hour, that is \(2 \times 250 \times e^{r}\). So, we can write the equation as:\(250 \times e^{10r} = 2 \times 250 \times e^{r}\)
02

Solve for the growth rate

Solve the equation from Step 1 to find the growth rate r. The equation simplifies to \(e^{9r} = 2\). Take the natural logarithm on both sides, we get \(9r = ln(2)\), so \(r = ln(2)/9.\)
03

Calculate the population after 6 hours

Use the growth rate found in Step 2 to find the population after 6 hours using the formula \(N = N_0 \times e^{rt}\) , we get \(N = 250 \times e^{6\times(ln(2)/9)}\)

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