Problem 24 - 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: \(500 \mathrm{mg}\) Time: after 3.5 hours

Step-by-Step Solution

Answer

Approximately 252.98 mg of aspirin is remaining in a person's bloodstream after 3.5 hours of taking a 500 mg dose.

Step 1: Identify given values

The initial amount of aspirin, \(A\), taken is given as 500 mg. The time, \(t\), since the medication was taken is 3.5 hours. These will be the values plugged into the equation to find the remaining amount.

Step 2: Substitute the given values

Put the designated values into the equation: \(y=500(0.8)^{3.5}\).

Step 3: Calculate the amount

Calculate the above equation to find the resulting amount of aspirin, \(y\), left in the person's bloodstream, which is approximately 252.982 mg.

Key Concepts

Algebra

Algebra is a fascinating branch of mathematics that allows us to manipulate equations and solve problems involving unknowns. In the context of exponential decay, algebra helps us understand how quantities decrease over time. In our exercise, we use an algebraic expression to model the decay of aspirin in the bloodstream. This model is expressed as the function:

• \( y = A(0.8)^t \)

where \( y \) is the remaining aspirin concentration, \( A \) is the initial dosage, and \( t \) is the time elapsed in hours.Since algebra deals with finding unknown values, it is crucial to know how to substitute the given data into the equation to find the desired quantity. Here, we substituted \( A = 500 \) mg and \( t = 3.5 \) hours into the equation to solve for \( y \). Always keep in mind that correctly identifying the values for substitution is a key part of algebra.

Mathematical Modeling

Mathematical modeling is the process of representing real-world situations using mathematical concepts. This makes it easier to analyze and solve complex problems. For the aspirin concentration scenario, we use mathematical modeling to predict how much of the medication remains in the bloodstream after a given time.The equation \( y = A(0.8)^t \) serves as a model for exponential decay. This means that the concentration of aspirin decreases exponentially as time progresses. Exponential decay models are common in nature, especially in processes related to medicine, radioactive decay, and depreciation of resources.One strength of mathematical models is their ability to generalize specific scenarios, giving us a reliable prediction tool for various conditions. Understanding the behavior of exponential functions is crucial here because it helps predict outcomes by analyzing the model's parameters, like the decay rate of 0.8 in our function.

Function Evaluation

Function evaluation involves calculating the output of a function for specific inputs. It is a basic yet significant concept in mathematics, and it is what allows us to use our mathematical model effectively.In our exercise, function evaluation is performed by substituting the known values into the model equation: \( y = 500(0.8)^{3.5} \). By evaluating this function, we find the remaining concentration of aspirin after 3.5 hours.To correctly evaluate a function, it is essential to follow the order of operations, sometimes remembered through the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In this example, after raising 0.8 to the power of 3.5, we multiply the result by 500 to get the final answer. These steps demonstrate how function evaluation turns an abstract concept into a tangible answer, enriching our understanding of the phenomenon.