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After 24 hours,
30% of a 25 milligram dose of a new antibiotic remains in the body. Which of the following functions,
M, models the amount of the antibiotic (in milligrams) that remains in the body after
h hours?
Choose 1 answer:
(A)
M(h)=30*(0.3)^((h)/( 24))
(B)
M(h)=25*(0.3)^((h)/( 24))
(c)
M(h)=25*(0.3)^(h)
(D)
M(h)=25*(0.7)^((h)/( 24))

After $24$ hours, $30 \%$ of a $25$ milligram dose of a new antibiotic remains in the body. Which of the following functions, $M$ , models the amount of the antibiotic (in milligrams) that remains in the body after $h$ hours? $\newline$ Choose $1$ answer: $\newline$ (A) $M(h)=30 \cdot(0.3)^{\frac{h}{24}}$ $\newline$ (B) $M(h)=25 \cdot(0.3)^{\frac{h}{24}}$ $\newline$ (C) $M(h)=25 \cdot(0.3)^{h}$ $\newline$ (D) $M(h)=25 \cdot(0.7)^{\frac{h}{24}}$

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Compare linear and exponential growth

Full solution

Q. After

24

hours,

30 \%

of a

25

milligram dose of a new antibiotic remains in the body. Which of the following functions,

M

, models the amount of the antibiotic (in milligrams) that remains in the body after

h

hours?

\newline

Choose

1

answer:

\newline

(A)

M(h)=30 \cdot(0.3)^{\frac{h}{24}}

\newline

(B)

M(h)=25 \cdot(0.3)^{\frac{h}{24}}

\newline

(C)

M(h)=25 \cdot(0.3)^{h}

\newline

(D)

M(h)=25 \cdot(0.7)^{\frac{h}{24}}

Identify Decay Model: We need to find a function that models the decay of the antibiotic in the body over time. We know that after $24$ hours, $30\%$ of the initial dose remains. This means that the amount of antibiotic decreases to $30\%$ of its initial amount every $24$ hours. We can use an exponential decay function to model this.
Exponential Decay Function: The general form of an exponential decay function is $M(h) = M_0 \times (\text{decay\_rate})^{(h / \text{time\_period})}$ , where $M_0$ is the initial amount, $\text{decay\_rate}$ is the percentage that remains after each time period, and $\text{time\_period}$ is the length of the time period after which the $\text{decay\_rate}$ is applied.
Substitute Values: In this case, $M_0$ is the initial dose, which is $25$ milligrams. The $\text{decay\_rate}$ is $30\%$ , or $0.3$ , since $30\%$ of the antibiotic remains after $24$ hours. The $\text{time\_period}$ is $24$ hours, as this is the period after which the $\text{decay\_rate}$ is applied.
Final Exponential Decay Function: Substituting these values into the general form of the exponential decay function, we get $M(h) = 25 \times (0.3)^{\frac{h}{24}}$ . This matches option (B) from the given choices.

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