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This exercise uses the exponential growth model.



A certain culture of the bacterium Rhodobacter sphaeroides initially has 10 bacteria and is observed to double every 4 hours.


AFind an exponential model $n(t)=n_0 2^{t / a}$ for the number of bacteria in the culture after $t$ hours.

$n(t)=$


BEstimate the number of bacteria after 17 hours. (Round your answer to the nearest whole number.)

______ bacteria


CAfter how many hours will the bacteria count reach 1 million? (Round your answer to one decimal place.)

$t=$

______ hr

Verified Answer :

  • The exponential model is defined as \(n(t) = 10 \cdot 2^{t/4}\).
  • After \(17\) hours, the population reaches approximately \(\boxed{190}\) bacteria.
  • The population reaches \(1\) million bacteria in \(\boxed{66.4}\) hours.

Explanation

  1. Defining the Exponential Growth Model

The exponential growth of the Rhodobacter sphaeroides bacteria population follows the standard form \(n(t) = n_0 2^{t/a}\), where:

  • \(n_0\) is the initial number of bacteria, given as \(10\).
  • \(a\) is the time it takes for the population to double, given as \(4\) hours.

By substituting these values into the general formula, the exponential model for the bacteria population at any time \(t\) (in hours) is:
\[n(t) = 10 \cdot 2^{t/4}\]

  1. Estimating Population After 17 Hours

To estimate the population after \(17\) hours, we substitute \(t = 17\) into our model:
\[n(17) = 10 \cdot 2^{17/4} = 10 \cdot 2^{4.25}\]
Calculating \(2^{4.25} \approx 19.0273\), we find:
\[n(17) = 10 \cdot 19.0273 \approx 190.27\]
Rounding to the nearest whole number, the number of bacteria after \(17\) hours is approximately \(190\).

  1. Calculating Time to Reach One Million Bacteria

To determine when the population reaches \(1\) million, we set the model equal to \(1,000,000\):
\[10 \cdot 2^{t/4} = 1,000,000\]
Divide both sides by \(10\):
\[2^{t/4} = 100,000\]
Next, take the logarithm (base 2) of both sides:
\[\frac{t}{4} = \log_2(100,000)\]
Using the change of base formula (\(\log_b(x) = \frac{\ln(x)}{\ln(b)}\)), we calculate:
\[t = 4 \cdot \frac{\ln(100,000)}{\ln(2)} \approx 4 \cdot 16.6096 \approx 66.4386\]
Rounding to one decimal place, the bacteria count reaches \(1\) million after \(66.4\) hours.

Examples

Exponential growth models like this are essential in fields like medicine and biology to understand how quickly diseases spread or how fast bacterial cultures grow in labs. For instance, in a controlled food manufacturing environment, identifying the exact time it takes for a microbial population to exceed safety thresholds allows food safety officers to set proper shelf-life expectations and implement sterilization protocols, ensuring that products remain safe for consumers. Understanding this model helps professionals predict future outcomes based on current growth rates and make informed, data-driven decisions.

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