Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Drug dosage A drug is eliminated from the body through urine. Suppose that for an initial dose of 10 milligrams, the amount \(A(t)\) in the body \(t\) hours later is given by \(A(t)=10(0.8)^{t}\) (a) Estimate the amount of the drug in the body 8 hours after the initial dose. (b) What percentage of the drug still in the body is eliminated each hour?

Step-by-Step Solution

Answer

(a) 1.68 mg. (b) 20% per hour.

Step 1: Understanding the Function

The given function for the amount of drug in the body is \(A(t) = 10(0.8)^t\). This shows an exponential decay where the initial dose is 10 mg and the drug decreases by a factor of 0.8 each hour.

Step 2: Calculate Amount After 8 Hours

To find the amount of drug in the body after 8 hours, substitute \(t = 8\) into the function: \[A(8) = 10(0.8)^8\] Calculate \((0.8)^8\): \((0.8)^8 \approx 0.16777216\), then, \[A(8) = 10 \times 0.16777216 \approx 1.68\text{ mg}\] Thus, the amount of drug left after 8 hours is approximately 1.68 mg.

Step 3: Determine Percentage of Drug Eliminated Each Hour

Each hour, the drug is multiplied by 0.8, meaning 20% is eliminated and 80% remains. To calculate the decay rate, subtract the remaining percentage from 100%: \[100\% - 80\% = 20\%\] Hence, 20% of the drug is eliminated each hour.

Key Concepts

Drug Elimination

Drug elimination is the process through which a drug is removed from the body, typically through bodily fluids such as urine. Understanding this concept is crucial, especially in pharmacology, to determine how long a drug stays active in the system. In the given scenario, the drug eliminated follows an exponential decay model. Here, it means that after every hour, a fixed percentage of the drug's remaining amount is cleared from the body. This predictable pattern helps medical professionals decide on appropriate dosages and intervals between doses to maintain therapeutic levels without causing harm.

Knowing the rate at which a drug is eliminated can be essential for avoiding overdoses and managing side effects. As shown, 20% of the drug is eliminated each hour, indicating a steady, predictable process. This knowledge guides how frequently doses are administered to keep the drug's concentration in the bloodstream at an effective level without exceeding safety thresholds.

Exponential Function

An exponential function is a mathematical expression that describes situations where a quantity decreases or increases at a constant rate per time unit. The general form is expressed as \( A(t) = A_0 imes r^t \), where \( A_0 \) denotes the initial amount, \( r \) is the decay or growth factor, and \( t \) references the time elapsed.

In the problem, the function \( A(t) = 10(0.8)^t \) represents exponential decay. Here, the initial dose is 10 mg, and the decay factor is 0.8. This means that each hour, the remaining amount is 80% of what it was the previous hour. The function allows us to estimate how much of the drug remains at any time \( t \).

These functions are prevalent in various fields due to their ability to model situations of change over time. For example, they not only represent drug elimination but also population growth and radioactive decay.

Percentage Calculation

Percentage calculation involves determining how much one quantity is relative to another, expressed as a percent. It plays a crucial role in understanding real-world applications, like the rate of drug elimination in the body.

In the given example, we are interested in how much of the drug is eliminated each hour. Since the problem states that each hour, the remaining drug amount is multiplied by 0.8, it means 80% remains, and thus 20% is eliminated. This calculation can be done by subtracting from 100%:

• Start with 100%, representing the total original amount.
• Subtract the percentage that remains, in this case, 80%.
• The result, 20%, shows how much is eliminated each hour.

This simple calculation shows how powerful percentage concepts can be when analyzing data, especially in the medical field where such measurements inform dosing regimens and treatment plans.