Problem 23 - Algebra 1

Question

Use the following information. The concentration of aspirin in a person's bloodstream can be modeled by the equation \(y=A(0.8)^{t},\) where \(y\) represents the concentration of aspirin in a person's bloodstream in milligrams (mg), \(A\) represents the amount of aspirin taken, and \(t\) represents the number of hours since the medication was taken. Find the amount of aspirin remaining in a person's bloodstream at the given dosage. Dosage: \(250 \mathrm{mg}\) Time: after 2 hours

Step-by-Step Solution

Answer

The amount of aspirin remaining in the person's bloodstream after 2 hours is approximately 160mg.

Step 1: Interpret the given variables

First, interpret what each variable in the equation represents: \(y\) represents the concentration of aspirin in a person's bloodstream in milligrams (mg), \(A\) represents the amount of aspirin taken, and \(t\) represents the number of hours since the medication was taken. In this problem, \(A=250mg\) (dosage) and \(t=2\) hours (time).

Step 2: Substitute the values into the equation

Next, replace the variables A and t with the given values in the model equation: \(y = A(0.8)^t\) becomes \(y = 250(0.8)^2\).

Step 3: Calculate the concentration

Calculate the value of \(y\), which represents the concentration of aspirin in the bloodstream after 2 hours. You compute this by multiplying 250 by 0.8 squared. This results in \(y \approx 160mg\).

Key Concepts

Concentration Modeling

Concentration modeling is a technique used to represent how the concentration of a substance changes over time. In pharmacology, it is essential for understanding how drugs work within the body, as it helps predict how long a drug will remain effective and when a next dose might be required. Here, the concentration model is given by the equation \(y = A(0.8)^{t}\). This equation shows exponential decay, a process where the concentration of aspirin decreases by a consistent percentage over equal time periods.

• \(y\): concentration of aspirin in mg
• \(A\): the initial dosage
• \(t\): time in hours after administration

The factor \(0.8\) in the equation signifies that approximately 80% of the drug remains in the bloodstream after each hour, indicating a decay factor of 0.8. It's important to understand that exponential decay is common in scenarios where the rate of a process is proportional to its current state, such as radioactive decay, cooling objects, or in this case, drug concentration in blood.

Variable Substitution

Variable substitution is a fundamental technique in solving mathematical problems, particularly in cases involving equations. It involves replacing variables in an equation with known quantities or other expressions to simplify the problem. This step allows us to convert a general model into a specific scenario to evaluate particular outcomes.
In the problem, we are given the values for \(A\) and \(t\):

• \(A = 250\) mg
• \(t = 2\) hours

By substituting these values into our concentration model \(y = A(0.8)^{t}\), the equation becomes \(y = 250(0.8)^2\). This substitution is critical as it helps transition from the abstract model to a concrete calculation, enabling the determination of the concentration after a specific period while using a specific dosage.

Equation Solving

Equation solving is the process of finding a value for the unknown variable that satisfies the equation. After substituting the known values into the exponential decay model, we arrive at the specific equation \(y = 250(0.8)^2\). Our goal here is to solve for \(y\), which represents the aspirin concentration in milligrams.
Let's break down the calculation step-by-step:

• First compute \((0.8)^2 = 0.64\). This represents the fraction of aspirin remaining after 2 hours.
• Multiply this result by 250, the original dosage: \(250 \times 0.64 = 160\).

Therefore, \(y = 160\) mg, meaning 160 mg of aspirin remains in the bloodstream after 2 hours. By using these steps of equation solving, you can accurately determine how drug concentrations decrease over time in the bloodstream.