After a special medicine is introduced into a petri dish full of bacteria, the number of bacteria remaining in the dish decreases rapidly.
The relationship between the elapsed time
\[t\], in seconds, and the number of bacteria,
\[B(t)\], in the petri dish is modeled by the following function:

\[B(t)=9300\cdot \left(\dfrac{1}{64}\right)^{t}\]
Complete the following sentence about the rate of change of the bacterial culture. Round your answer to two decimal places.
The bacterial culture loses

\[\dfrac 12\] of its size every
seconds.

The function provided to model the number of bacteria in the petri dish is:

\[
B(t) = 9300 \cdot \left(\dfrac{1}{64}\right)^t
\]

This is an exponential decay function. To find how long it takes for the bacterial culture to lose \(\dfrac{1}{2}\) of its size, we need to determine the time \(t\) at which the bacteria count is halved.

Let's find the half-life, i.e., the time \(t_{1/2}\) such that:

\[
B(t_{1/2}) = \dfrac{B(0)}{2}
\]

Since \(B(0) = 9300\), we set:

\[
9300 \cdot \left(\dfrac{1}{64}\right)^{t_{1/2}} = \dfrac{9300}{2}
\]

Simplifying, we get:

\[
\left(\dfrac{1}{64}\right)^{t_{1/2}} = \dfrac{1}{2}
\]

Now, let's solve for \(t_{1/2}\):

Take the natural logarithm (logarithm base \(e\)) of both sides:

\[
\ln\left(\left(\dfrac{1}{64}\right)^{t_{1/2}}\right) = \ln\left(\dfrac{1}{2}\right)
\]

Using the property \(\ln(a^b) = b \ln(a)\):

\[
t_{1/2} \cdot \ln\left(\dfrac{1}{64}\right) = \ln\left(\dfrac{1}{2}\right)
\]

Solve for \(t_{1/2}\):

\[
t_{1/2} = \dfrac{\ln\left(\dfrac{1}{2}\right)}{\ln\left(\dfrac{1}{64}\right)}
\]

Now calculate:

\[
\ln\left(\dfrac{1}{2}\right) \approx -0.6931
\]
\[
\ln\left(\dfrac{1}{64}\right) = \ln\left(64^{-1}\right) = -\ln(64) \approx -\ln(2^6) = -6\ln(2) \approx -6 \times 0.6931 = -4.1587
\]

Finally, substitute and calculate:

\[
t_{1/2} = \dfrac{-0.6931}{-4.1587} \approx 0.17 \text{ seconds}
\]

Thus, the bacterial culture loses \(\dfrac{1}{2}\) of its size every 0.17 seconds.

---

Do you want more details on this process or have any questions?

### Related Questions:

1. What is the significance of the base in the exponential decay function?
2. How can you use logarithms to solve other exponential equations?
3. What does the initial value 9300 represent in the context of this problem?
4. How would the function change if the medicine had a slower effect?
5. How do you interpret the negative sign in the logarithmic calculation?
6. What would the bacterial count be after 1 second?
7. Can this function be used to model other types of decay, such as radioactive decay?
8. How does the value of \( \dfrac{1}{64} \) influence the rate of bacterial decrease?

### Tip:
When dealing with exponential functions, understanding the base and its relationship to time is crucial for interpreting growth or decay rates.

Explore how exponential decay models the reduction of bacterial culture over time. Learn about the relationship between elapsed time and the number of bacteria remaining in a petri dish. Find out how quickly the bacterial culture loses half of its size. Detailed solution with step-by-step calculations.

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