After a certain medicine is ingested, the number of harmful bacteria remaining in the body declines rapidly.

The relationship between the elapsed time [tex]t[/tex], in minutes, since the medicine is ingested, and the number of harmful bacteria remaining in the body, [tex]H_{\text{minute}}(t)[/tex], is modeled by the following function:

[tex]H_{\text{minute}}(t) = 500,000,000 \cdot (0.2)^t[/tex]

Complete the following sentence about the rate of change in the number of harmful bacteria remaining in the body in seconds. Round your answer to two decimal places.

Every second, the number of harmful bacteria remaining in the body decays by a factor of [tex]\square[/tex].

We start with the information that after one minute (60 seconds) the bacteria reduce by a factor of [tex]$0.2$[/tex]. This means that if [tex]$d$[/tex] is the decay factor per second, then after 60 seconds the combined decay is

[tex]$$
d^{60} = 0.2.
$$[/tex]

To find [tex]$d$[/tex], we take the 60th root of both sides:

[tex]$$
d = (0.2)^{\frac{1}{60}}.
$$[/tex]

Calculating [tex]$(0.2)^{\frac{1}{60}}$[/tex] gives approximately

[tex]$$
d \approx 0.9735326020510389.
$$[/tex]

When rounded to two decimal places, we obtain

[tex]$$
d \approx 0.97.
$$[/tex]

Thus, every second, the number of harmful bacteria in the body decays by a factor of [tex]$0.97$[/tex].