Questions: The count in a bacteria culture was 300 after 15 minutes and 1000 after 35 minutes. Assuming the count grows exponentially, What was the initial size of the culture? Find the doubling period. Find the population after 75 minutes. When will the population reach 10000. You may enter the exact value or round to 2 decimal places.

The count in a bacteria culture was 300 after 15 minutes and 1000 after 35 minutes. Assuming the count grows exponentially,

What was the initial size of the culture?

Find the doubling period.

Find the population after 75 minutes.

When will the population reach 10000.

You may enter the exact value or round to 2 decimal places.
Transcript text: The count in a bacteria culture was 300 after 15 minutes and 1000 after 35 minutes. Assuming the count grows exponentially, What was the initial size of the culture? $\square$ Find the doubling period. $\square$ Find the population after 75 minutes. $\square$ When will the population reach 10000. $\square$ You may enter the exact value or round to 2 decimal places.
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Solution

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Solution Steps

Hint

To solve exponential growth problems, first determine the growth rate using given data points and the exponential growth formula. Then, use this rate to find the initial population size, doubling period, future population at a specific time, and the time it will take to reach a certain population size.

Step 1: Determine the Initial Size of the Culture

Using the exponential growth formula P(t)=P0ekt P(t) = P_0 e^{kt} , we have two data points:

  1. P1=300 P_1 = 300 at t1=15 t_1 = 15
  2. P2=1000 P_2 = 1000 at t2=35 t_2 = 35

By dividing the equations, we eliminate P0 P_0 : 1000300=e20k    103=e20k \frac{1000}{300} = e^{20k} \implies \frac{10}{3} = e^{20k} Taking the natural logarithm: ln(103)=20k    k=ln(103)200.060198640216296805 \ln\left(\frac{10}{3}\right) = 20k \implies k = \frac{\ln\left(\frac{10}{3}\right)}{20} \approx 0.060198640216296805

Now, substituting k k back into the equation for P1 P_1 : P0=P1ekt1=300e15k121.6080139326331 P_0 = \frac{P_1}{e^{kt_1}} = \frac{300}{e^{15k}} \approx 121.6080139326331 Thus, the initial size of the culture is: P0121.61 \boxed{P_0 \approx 121.61}

Step 2: Find the Doubling Period

The doubling period T T can be calculated using the formula: T=ln(2)k11.514332849868898 T = \frac{\ln(2)}{k} \approx 11.514332849868898 Thus, the doubling period is: T11.51 minutes \boxed{T \approx 11.51 \text{ minutes}}

Step 3: Population After 75 Minutes

To find the population after t3=75 t_3 = 75 minutes, we use: P(75)=P0ek7511111.111111111117 P(75) = P_0 e^{k \cdot 75} \approx 11111.111111111117 Thus, the population after 75 minutes is: P(75)11111.11 \boxed{P(75) \approx 11111.11}

Final Answer

  • Initial size of the culture: 121.61 \boxed{121.61}
  • Doubling period: 11.51 minutes \boxed{11.51 \text{ minutes}}
  • Population after 75 minutes: 11111.11 \boxed{11111.11}
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