Question

Math

Posted 4 months ago

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A pharmaceutical scientist making a new medication wonders how much of the active ingredient to include in a dose. They are curious how long different amounts of the active ingredient will stay in someone's bloodstream.
The amount of time (in hours) the active ingredient remains in the bloodstream can be modeled by $f(x)=-1.25 \cdot \ln \left(\frac{1}{x}\right)$, where $x$ is the initial amount of the active ingredient (in milligrams). Here is the graph of $f$ and the graph of the line $y=4$.
Which statements represent the meaning of the intersection point of the graphs?
Choose all answers that apply:
A It describes the amount of time the active ingredient stays in the bloodstream if the initial amount of the active ingredient is $4 \mathrm{mg}$.
B It gives the solution to the equation $-1.25 \cdot \ln \left(\frac{1}{x}\right)=4$.
c. It describes the situation where the initial amount of the active ingredient is equal to how long it stays in the bloodstream.
D It gives the initial amount of the active ingredient such that the last of the active ingredient leaves the bloodstream after 4 hours.
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Answer from Sia

Posted 4 months ago

Solution by Steps

step 2

The intersection point occurs where $f(x) = y$. Therefore, at the intersection, $-1.25 \cdot \ln \left(\frac{1}{x}\right) = 4$

step 3

To solve for $x$, we can use the property of logarithms that $\ln(a) = b$ implies $e^b = a$

step 4

Applying this property to our equation, we get $e^{-\frac{4}{1.25}} = \frac{1}{x}$, which simplifies to $x = e^{\frac{16}{5}}$

step 5

The value of $x$ represents the initial amount of the active ingredient in milligrams such that the active ingredient stays in the bloodstream for 4 hours

[question number] Answer

B

Key Concept

Intersection Point of Graphs

Explanation

The intersection point of the graph of a function and a horizontal line represents the solution to the equation formed by setting the function equal to the y-value of the horizontal line.

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