Problem 26 - Algebra 1: Concepts and Skills

Question

In Exercises 26–28, use the following information. The amount of aspirin y (in milligrams) in a person’s blood can be modeled by y A(0.8)t where A represents the dose of aspirin taken (in milligrams) and t represents the number of hours since the aspirin was taken. Find the amount of aspirin remaining in a person’s blood for the given dosage and time. $$Dosage: 250 \mathrm{mg}\quad Time: after 2 hours$$

Step-by-Step Solution

Answer

The amount of aspirin remaining in a person's blood after 2 hours is 160 mg.

Step 1: Identify variables

Identify what each variable in the equation corresponds to. In this case, A is the dose taken, which is 250 mg, and t is the number of hours since the aspirin was taken, which is 2.

Step 2: Substitute the variables

Substitute the values A = 250 mg and t = 2 hours into the equation: y = 250*(0.8)^2.

Step 3: Perform the Calculations

Calculate y by first raising 0.8 to the power of 2 to get 0.64, and then multiplying the result by 250 to get y = 160 mg.

Key Concepts

Mathematical Modeling

Mathematical modeling is a way of representing real-world situations using mathematical concepts and language. In the exercise, mathematical modeling is used to predict the concentration of aspirin in the blood. This is achieved through an exponential decay formula.
This kind of modeling involves:

• Identifying variables: Understanding what each symbol represents in the equation.
• Formulating equations: Creating an equation that closely reflects the real-life situation.
• Solving the equation: Calculating the values that satisfy the conditions of the problem.

In the given exercise, the equation is set up to model how aspirin breaks down in the bloodstream over time. The variable A stands for the initial dosage, and t represents the time that has passed since ingestion, which helps in finding how much aspirin remains.

Substitution Method

The substitution method is a key step in solving equations, especially when variables need to be replaced by their respective numerical values. In this exercise, the equation involves two variables: A (the dosage of aspirin) and t (the time in hours).
When solving the problem:

• First, it's crucial to correctly identify the given variables and their respective values, as done in Step 1 of the solution.
• Then, substitute these values into the equation: for A, use 250 mg; for t, use 2 hours.
• This transforms the abstract equation into a concrete one, which is easier to handle mathematically.

By substituting the values, the expression becomes manageable and allows for straightforward computation of the remaining aspirin amount.

Exponential Functions

Exponential functions play a significant role in modeling processes that involve growth or decay. An exponential function includes a constant base raised to a variable exponent. In this problem, the base is 0.8, representing the rate of decay, while the exponent is t, or time elapsed.
Exponential decay functions follow the formula:

• The general form: \(y = A(b)^t\), where \(b\) is the decay factor (0.8 here) reflecting the percentage of substance that remains after each time unit.
• As time increases, the base raised to the power of the time diminishes, demonstrating decay. This is typical in situations such as radioactive decay or medication breakdown.

In the exercise, after substituting the values, compute \(0.8^2 = 0.64\), meaning only 64% of the aspirin remains per hour passed. Multiply by 250mg to find the exact remaining amount, which is 160mg after 2 hours. This highlights how exponential functions are effectively used to model such scenarios.