The amount of medication in Rory's bloodstream decreases at a rate that is proportional at any time to the amount of the medication in the bloodstream at that time. $\newline$ Rory takes $150$ milligrams of medication initially. The amount of medication is halved every $13$ hours. $\newline$ How many milligrams of the medication are in Rory's bloodstream after $8$ hours? $\newline$ Choose $1$ answer: $\newline$ (A) $98$ $\newline$ (B) $\mathbf{1 0 4}$ $\newline$ (C) $204$

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Q. The amount of medication in Rory's bloodstream decreases at a rate that is proportional at any time to the amount of the medication in the bloodstream at that time.

\newline

Rory takes

150

milligrams of medication initially. The amount of medication is halved every

13

hours.

\newline

How many milligrams of the medication are in Rory's bloodstream after

8

hours?

\newline

Choose

1

answer:

\newline

(A)

98

\newline

(B)

\mathbf{1 0 4}

\newline

(C)

204

Understand the problem: Understand the problem. $\newline$ We are given that the amount of medication in Rory's bloodstream decreases by half every $13$ hours. This is an example of exponential decay, which can be described by the formula $A = A_0 \times (1/2)^{(t/T)}$ , where $A$ is the amount of medication at time $t$ , $A_0$ is the initial amount of medication, $T$ is the half-life of the medication, and $t$ is the time elapsed.
Identify the known values: Identify the known values. $\newline$ The initial amount of medication $A_0$ is $150$ milligrams, the half-life $T$ is $13$ hours, and the time elapsed $t$ is $8$ hours.
Substitute values into formula: Substitute the known values into the exponential decay formula. $A = 150 \times \left(\frac{1}{2}\right)^{\frac{8}{13}}$
Calculate the exponent: Calculate the exponent. $\newline$ $(\frac{8}{13})$ is approximately $0.6154$ (rounded to four decimal places).
Calculate amount after $8$ hours: Calculate the amount of medication after $8$ hours. $\newline$ $A = 150 \times (1/2)^{0.6154}$ $\newline$ To calculate $(1/2)^{0.6154}$ , we can use a calculator.
Perform the calculation: Perform the calculation. $\newline$ $(1/2)^{0.6154}$ is approximately $0.6924$ (rounded to four decimal places). $\newline$ Now, multiply this by the initial amount of medication: $\newline$ $A = 150 \times 0.6924$ $\newline$ $A \approx 103.86$ milligrams
Round the answer: Round the answer to the nearest whole number, as medication is typically measured in whole milligrams. $\newline$ $A \approx 104$ milligrams