After a special medicine is introduced into a petri dish full of bacteria, the number of bacteria remaining in the dish decreases rapidly. $\newline$ The relationship between the elapsed time $t$ , in seconds, and the number of bacteria, $B_{\text {second }}(t)$ , in the petri dish is modeled by the following function: $\newline$ $B_{\text {second }}(t)=6000 \cdot\left(\frac{15}{16}\right)^{t}$ $\newline$ Complete the following sentence about the rate of change in the number of bacteria in minutes. Round your answer to two decimal places. $\newline$ Every minute, the number of bacteria decays by a factor of

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Q. After a special medicine is introduced into a petri dish full of bacteria, the number of bacteria remaining in the dish decreases rapidly.

\newline

The relationship between the elapsed time

t

, in seconds, and the number of bacteria,

B_{\text {second }}(t)

, in the petri dish is modeled by the following function:

\newline

B_{\text {second }}(t)=6000 \cdot\left(\frac{15}{16}\right)^{t}

\newline

Complete the following sentence about the rate of change in the number of bacteria in minutes. Round your answer to two decimal places.

\newline

Every minute, the number of bacteria decays by a factor of

Understand function: Understand the given function and what it represents. $\newline$ The function $B_{\text{second}}(t)=6000\left(\frac{15}{16}\right)^t$ models the number of bacteria in the petri dish at any given time $t$ in seconds. We need to find the rate of change in the number of bacteria in minutes, not seconds.
Convert time to minutes: Convert the time from seconds to minutes. $\newline$ Since there are $60$ seconds in a minute, we need to find the decay factor for $60$ seconds to determine the decay factor per minute. We will replace $t$ with $60$ in the given function.
Calculate bacteria after $1$ minute: Calculate the number of bacteria after $1$ minute. $B_{\text{minute }}(1) = 6000\left(\frac{15}{16}\right)^{60}$
Calculate decay factor for $1$ minute: Perform the calculation for the number of bacteria after $1$ minute. $\newline$ $B_{\text{minute}}(1) = 6000\times\left(\frac{15}{16}\right)^{60}$ $\newline$ To simplify the calculation, we can first calculate the decay factor $\left(\frac{15}{16}\right)^{60}$ .
Multiply by decay factor: Calculate the decay factor for $1$ minute. $\left(\frac{15}{16}\right)^{60} = (0.9375)^{60}$ Using a calculator, we find that $(0.9375)^{60} \approx 0.177978515625$ .
Determine decay factor: Multiply the initial number of bacteria by the decay factor to find the number of bacteria after $1$ minute. $\newline$ $B_{\text{minute }}(1) = 6000 \times 0.177978515625$ $\newline$ $B_{\text{minute }}(1) \approx 1067.87109375$
Round decay factor: Determine the decay factor by comparing the number of bacteria after $1$ minute to the initial number of bacteria. $\newline$ The decay factor is the ratio of the number of bacteria after $1$ minute to the initial number of bacteria. $\newline$ Decay factor = $B_{\text{minute }}(1) / B_{\text{second }}(0)$ $\newline$ Decay factor $\approx 1067.87109375 / 6000$ $\newline$ Decay factor $\approx 0.177978515625$
Round decay factor: Determine the decay factor by comparing the number of bacteria after $1$ minute to the initial number of bacteria. $\newline$ The decay factor is the ratio of the number of bacteria after $1$ minute to the initial number of bacteria. $\newline$ Decay factor = $B_{\text{minute }}(1) / B_{\text{second }}(0)$ $\newline$ Decay factor $\approx 1067.87109375 / 6000$ $\newline$ Decay factor $\approx 0.177978515625$ Round the decay factor to two decimal places. $\newline$ Decay factor $\approx 0.18$