Problem 55 - Algebra and Trigonometry with Analytic Geometry

Question

If a 100 -milligram tablet of an asthma drug is taken orally and if none of the drug is present in the body when the tablet is first taken, the total amount \(A\) in the bloodstream after \(t\) minutes is predicted to be $$ A=100\left[1-(0.9)^{t}\right] \quad \text { for } \quad 0 \leq t \leq 10 $$ (a) Sketch the graph of the equation. (b) Determine the number of minutes needed for \(50 \mathrm{mil}-\) ligrams of the drug to have entered the bloodstream.

Step-by-Step Solution

Answer

Approximately 6.57 minutes for 50 mg to enter the bloodstream.

Step 1: Understand the Formula

The equation given is \( A = 100 [1-(0.9)^t] \). It represents the amount of drug in the bloodstream over time, where \( A \) is in milligrams and \( t \) is time in minutes. The initial condition is that the tablet is taken at \( t = 0 \) and no drug is initially present.

Step 2: Sketch the Graph

To sketch the graph of the function \( A = 100 [1-(0.9)^t] \), note that as \( t \) increases from 0 to 10, \( (0.9)^t \) decreases, so \( 1-(0.9)^t \) increases. At \( t = 0 \), \( A = 100 [1 - 1] = 0 \). As \( t \) approaches 10, \( (0.9)^t \) approaches 0, making \( A \) approach 100 mg. So the graph starts at (0,0) and rises towards (10,100), showing an exponential growth curve.

Step 3: Set Up the Equation for Part (b)

To find the time \( t \) when 50 milligrams are in the bloodstream, set \( A = 50 \): \[ 50 = 100 [1-(0.9)^t] \]. Simplify this to find \( t \).

Step 4: Solve for t

First divide both sides by 100: \( 0.5 = 1-(0.9)^t \). Rearranging gives \( (0.9)^t = 0.5 \). Now take the logarithm of both sides to solve for \( t \): \( t = \frac{\log(0.5)}{\log(0.9)} \).

Step 5: Calculate Value of t

Using a calculator, compute \( \log(0.5) \approx -0.3010 \) and \( \log(0.9) \approx -0.0458 \). Thus, \( t \approx \frac{-0.3010}{-0.0458} \approx 6.57 \). Therefore, it takes approximately 6.57 minutes for 50 milligrams of the drug to enter the bloodstream.

Key Concepts

Drug Concentration Modeling

Understanding drug concentration modeling can help predict how a drug behaves in the body over time. In this exercise, we look at how a 100-milligram asthma drug disperses in the bloodstream. This is a classic example of using exponential functions to describe natural processes. The formula, \( A = 100 \left[1-(0.9)^t\right] \), helps us compute the concentration \( A \) in the bloodstream at any time \( t \).
Here, as time progresses, the factor \( (0.9)^t \) decreases, indicating a reduction in the proportion of unused drug. Initially, when \( t = 0 \), the drug concentration is zero. As \( t \) approaches 10 minutes, the drug concentration rises towards 100 mg. This model helps in understanding how rapidly the drug enters the bloodstream, which is crucial for ensuring patient safety and effective treatment.
Such models are instrumental for healthcare professionals to ensure that medication is given at the right dosage and interval to maintain an effective therapeutic level.

Graph Sketching

Graph sketching is a valuable skill for visualizing functions like the drug concentration model. The function \( A = 100 [1-(0.9)^t] \) describes an exponential curve. To sketch this graph, consider how the components change as \( t \) ranges from 0 to 10.
Your starting point is \( t = 0 \), where \( A = 0 \) mg, meaning no drug has entered the bloodstream. As \( t \) increases, \( (0.9)^t \) decreases, causing \( A \) to increase. By \( t = 10 \), \( A \) reaches close to 100 mg. The graph therefore begins at (0,0) and rises towards (10,100), reflecting an exponential increase.

• Start by plotting key points: (0,0), (5, ~58), and (10, ~100).
• The curve should gently rise, steeper initially, and flattening as \( t \) reaches 10.

This graph offers a quick, visual summarization of the drug's transition from non-existent in the body to fully absorbed.

Solving Logarithmic Equations

Solving logarithmic equations often involves isolating terms and using properties of logarithms. For this exercise, we're determining how long it takes for 50 milligrams to be in the bloodstream. Start with the equation: \( 50 = 100 \left[1-(0.9)^t\right] \).
Simplify by dividing both sides by 100 to get \( 0.5 = 1 - (0.9)^t \). Then, rearrange to \( (0.9)^t = 0.5 \). To eliminate the exponential term, take the logarithm of both sides:
\[ t = \frac{\log(0.5)}{\log(0.9)} \]
Calculating gives approximately \( t = 6.57 \) minutes. This solution involves clear steps:

• Rearrange the equation to isolate the exponential term.
• Apply logarithms to solve for \( t \).
• Use a calculator for accurate logarithmic computation.

Mastering this technique enables you to solve real-world problems where changes are modeled exponentially, such as drug absorption rates in the body.