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f(m)=500((1)/(2))^((m)/( 20))
The function models
f, the amount of a particular medicine in milligrams in a patient's bloodstream,
m minutes after the medicine is fully absorbed. Based on the function, how many minutes does it take for the amount of medicine in the patient's bloodstream to reduce by half?
Choose 1 answer:
(A)
(1)/(2)
(B) 10
(c) 20
(D) 250

$f(m)=500\left(\frac{1}{2}\right)^{\frac{m}{20}}$ $\newline$ The function models $f$ , the amount of a particular medicine in milligrams in a patient's bloodstream, $m$ minutes after the medicine is fully absorbed. Based on the function, how many minutes does it take for the amount of medicine in the patient's bloodstream to reduce by half? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{1}{2}$ $\newline$ (B) $\mathbf{1 0}$ $\newline$ (C) $20$ $\newline$ (D) $\mathbf{2 5 0}$

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Compare linear and exponential growth

Full solution

Q.

f(m)=500\left(\frac{1}{2}\right)^{\frac{m}{20}}

\newline

The function models

f

, the amount of a particular medicine in milligrams in a patient's bloodstream,

m

minutes after the medicine is fully absorbed. Based on the function, how many minutes does it take for the amount of medicine in the patient's bloodstream to reduce by half?

\newline

Choose

1

answer:

\newline

(A)

\frac{1}{2}

\newline

(B)

\mathbf{1 0}

\newline

(C)

20

\newline

(D)

\mathbf{2 5 0}

Given function: We are given the function $f(m) = 500 \times \left(\frac{1}{2}\right)^{\frac{m}{20}}$ , which models the amount of medicine in a patient's bloodstream $m$ minutes after the medicine is fully absorbed. We want to find the value of $m$ that makes $f(m)$ half of the initial amount, which is $500$ mg.
Finding half of initial amount: To reduce the amount of medicine by half, we need to find $m$ such that $f(m) = 500 \times (1/2)$ . This means we are looking for when $f(m)$ equals $250$ mg, since $500 \times (1/2) = 250$ .
Setting up the equation: Setting up the equation, we have $500 \times \left(\frac{1}{2}\right)^{\frac{m}{20}} = 250$ . To solve for $m$ , we can divide both sides of the equation by $500$ to isolate the exponential term.
Simplifying the equation: After dividing by $500$ , we get $(1/2)^{(m/20)} = 250/500$ , which simplifies to $(1/2)^{(m/20)} = 1/2$ .
Setting exponents equal: Since the bases are the same on both sides of the equation, we can set the exponents equal to each other. This gives us $\frac{m}{20} = 1$ .
Solving for m: Multiplying both sides of the equation by $20$ to solve for $m$ , we get $m = 20 \times 1$ , which simplifies to $m = 20$ .

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