Questions: The model that describes the number of bacteria in a culture after t days has just been updated from P(t)=15 * 3^t to P(t)=15 * 4^t. What implications can you draw from this information? - the initial number of bacteria is 4 instead of 3 - the number of bacteria is 4 times as much as the initial value instead of 3 times as much - the growth rate of bacteria is 40% per day instead of 30% per day - for each day that passes, the number of bacteria increases by 15 - the number of bacteria increases by 4 times a day instead of 3 times a day

The model that describes the number of bacteria in a culture after t days has just been updated from P(t)=15 * 3^t to P(t)=15 * 4^t. What implications can you draw from this information? - the initial number of bacteria is 4 instead of 3 - the number of bacteria is 4 times as much as the initial value instead of 3 times as much - the growth rate of bacteria is 40% per day instead of 30% per day - for each day that passes, the number of bacteria increases by 15 - the number of bacteria increases by 4 times a day instead of 3 times a day
Transcript text: The model that describes the number of bacteria in a culture after $t$ days has just been updated from $P(t)=15 \cdot 3^{t}$ to $P(t)=15 \cdot 4^{t}$. What implications can you draw from this information? the initial number of bacteria is 4 instead of 3 the number of bacteria is 4 times as much as the initial value instead of 3 times as much the growth rate of bacteria is $40 \%$ per day instead of $30 \%$ per day for each day that passes, the number of bacteria increases by 15 the number of bacteria increases by 4 times a day instead of 3 times a day
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Solution

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Solution Steps

Step 1: Analyze the original and updated models

The original model is \( P(t) = 15 \cdot 3^{t} \), and the updated model is \( P(t) = 15 \cdot 4^{t} \). Both models describe the number of bacteria in a culture after \( t \) days. The base of the exponential function has changed from 3 to 4.

Step 2: Interpret the change in the base

The base of the exponential function represents the growth factor. In the original model, the growth factor is 3, meaning the number of bacteria triples every day. In the updated model, the growth factor is 4, meaning the number of bacteria quadruples every day.

Step 3: Evaluate the implications
  • The initial number of bacteria is still 15 in both models, so the first option is incorrect.
  • The growth rate has increased from tripling to quadrupling, so the second option is correct.
  • The growth rate is not expressed as a percentage in the models, so the third option is incorrect.
  • The number of bacteria does not increase by 15 each day; it increases by a factor of 4, so the fourth option is incorrect.
  • The number of bacteria increases by 4 times a day instead of 3 times a day, so the fifth option is correct.

Final Answer

The correct implications are:

  • The number of bacteria is 4 times as much as the initial value instead of 3 times as much.
  • The number of bacteria increases by 4 times a day instead of 3 times a day.

\\(\boxed{\text{The correct options are the second and fifth.}}\\)

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