Answer :
We start with the function that models the number of bacteria remaining after the medicine is ingested:
[tex]$$
H_{\text{minute}}(t) = 500\,000\,000 \cdot (0.2)^t,
$$[/tex]
where [tex]$t$[/tex] is in minutes. This tells us that every minute, the number of harmful bacteria is multiplied by the factor [tex]$0.2$[/tex], meaning that there is a decay factor of [tex]$0.2$[/tex] per minute.
Since there are [tex]$60$[/tex] seconds in one minute, let [tex]$r$[/tex] be the decay factor per second. After [tex]$60$[/tex] seconds (which is one minute), the bacteria count is multiplied by [tex]$r^{60}$[/tex]. Because this must equal the one-minute decay factor, we equate:
[tex]$$
r^{60} = 0.2.
$$[/tex]
To find [tex]$r$[/tex], we take the [tex]$60^\text{th}$[/tex] root of [tex]$0.2$[/tex]:
[tex]$$
r = 0.2^{1/60}.
$$[/tex]
Calculating [tex]$0.2^{1/60}$[/tex] gives approximately [tex]$0.9735326020510389$[/tex]. Rounding this value to two decimal places, we obtain:
[tex]$$
r \approx 0.97.
$$[/tex]
Thus, every second, the number of harmful bacteria remaining in the body decays by a factor of
[tex]$$
\boxed{0.97}.
$$[/tex]
[tex]$$
H_{\text{minute}}(t) = 500\,000\,000 \cdot (0.2)^t,
$$[/tex]
where [tex]$t$[/tex] is in minutes. This tells us that every minute, the number of harmful bacteria is multiplied by the factor [tex]$0.2$[/tex], meaning that there is a decay factor of [tex]$0.2$[/tex] per minute.
Since there are [tex]$60$[/tex] seconds in one minute, let [tex]$r$[/tex] be the decay factor per second. After [tex]$60$[/tex] seconds (which is one minute), the bacteria count is multiplied by [tex]$r^{60}$[/tex]. Because this must equal the one-minute decay factor, we equate:
[tex]$$
r^{60} = 0.2.
$$[/tex]
To find [tex]$r$[/tex], we take the [tex]$60^\text{th}$[/tex] root of [tex]$0.2$[/tex]:
[tex]$$
r = 0.2^{1/60}.
$$[/tex]
Calculating [tex]$0.2^{1/60}$[/tex] gives approximately [tex]$0.9735326020510389$[/tex]. Rounding this value to two decimal places, we obtain:
[tex]$$
r \approx 0.97.
$$[/tex]
Thus, every second, the number of harmful bacteria remaining in the body decays by a factor of
[tex]$$
\boxed{0.97}.
$$[/tex]
