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The quantity of caffeine in the body after drinking a cup of coffee is shown in the figure to the right. The half-life is (a) Estimate the half-life of caffeine. NOTE: Round your answer to one decimal place. 150 hours. The continuous decay rate is 100 50 0 Q (mg) 4 8 (b) Use the half-life to find the continuous percent decay rate. NOTE: Round your answer to one decimal place. Remember to express your answer as a decay rate, not a growth rate. % per 12 t (hours) hour. (c) Give a formula for Q as a function of t. Assume a continuous decay rate. NOTE: Round the parameters in your answer to three decimal places. Q(t)

The quantity of caffeine in the body after drinking a cup of coffee is shown in the figure to the right. The half-life is (a) Estimate the half-life of caffeine. NOTE: Round your answer to one decimal place. 150 hours. The continuous decay rate is 100 50 0 Q (mg) 4 8 (b) Use the half-life to find the continuous percent decay rate. NOTE: Round your answer to one decimal place. Remember to express your answer as a decay rate, not a growth rate. % per 12 t (hours) hour. (c) Give a formula for Q as a function of t. Assume a continuous decay rate. NOTE: Round the parameters in your answer to three decimal places. Q(t)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Title: Understanding Caffeine Decay in the Human Body** --- **Introduction** When you drink a cup of coffee, the caffeine content in your body decreases over time. This process can be understood through concepts like half-life and decay rate. Let’s explore these concepts by analyzing the caffeine decay using a mathematical approach. --- **Graph Explanation** The graph on the right illustrates the caffeine quantity \( Q \) (in mg) in the body over time \( t \) (in hours). The curve shows a decreasing trend, demonstrating how caffeine levels diminish after coffee consumption. Initially, at \( t = 0 \), the quantity is approximately 150 mg. Over time, this amount decreases as shown by the downward slope of the graph. The grid lines on the graph can assist in estimating values at specific time intervals. --- **Tasks** **(a) Estimate the Half-Life of Caffeine** *Half-life* is the time required for the caffeine quantity to reduce to half its initial amount. Examine the graph to estimate this duration. *Note:* Round your answer to one decimal place. - **The half-life is \_\_\_\_ hours.** **(b) Calculate the Continuous Percent Decay Rate** Using the estimated half-life, determine the continuous percent decay rate. This rate indicates how quickly the caffeine amount decreases continuously over time. *Note:* Round your answer to one decimal place and express it as a decay rate. - **The continuous decay rate is \_\_\_\_ % per hour.** **(c) Develop a Formula for \( Q \) as a Function of \( t \)** Formulate an equation representing \( Q(t) \), the caffeine quantity at time \( t \), assuming a consistent continuous decay rate. *Note:* Round the parameters to three decimal places. - **\( Q(t) = \_\_\_\_ \)** --- **Conclusion** Understanding the caffeine decay rate and half-life is crucial for managing caffeine intake and its effects on the body. By utilizing mathematical models, such as exponential decay, we can visualize and calculate changes in caffeine concentration over time.
Transcribed Image Text:**Title: Understanding Caffeine Decay in the Human Body** --- **Introduction** When you drink a cup of coffee, the caffeine content in your body decreases over time. This process can be understood through concepts like half-life and decay rate. Let’s explore these concepts by analyzing the caffeine decay using a mathematical approach. --- **Graph Explanation** The graph on the right illustrates the caffeine quantity \( Q \) (in mg) in the body over time \( t \) (in hours). The curve shows a decreasing trend, demonstrating how caffeine levels diminish after coffee consumption. Initially, at \( t = 0 \), the quantity is approximately 150 mg. Over time, this amount decreases as shown by the downward slope of the graph. The grid lines on the graph can assist in estimating values at specific time intervals. --- **Tasks** **(a) Estimate the Half-Life of Caffeine** *Half-life* is the time required for the caffeine quantity to reduce to half its initial amount. Examine the graph to estimate this duration. *Note:* Round your answer to one decimal place. - **The half-life is \_\_\_\_ hours.** **(b) Calculate the Continuous Percent Decay Rate** Using the estimated half-life, determine the continuous percent decay rate. This rate indicates how quickly the caffeine amount decreases continuously over time. *Note:* Round your answer to one decimal place and express it as a decay rate. - **The continuous decay rate is \_\_\_\_ % per hour.** **(c) Develop a Formula for \( Q \) as a Function of \( t \)** Formulate an equation representing \( Q(t) \), the caffeine quantity at time \( t \), assuming a consistent continuous decay rate. *Note:* Round the parameters to three decimal places. - **\( Q(t) = \_\_\_\_ \)** --- **Conclusion** Understanding the caffeine decay rate and half-life is crucial for managing caffeine intake and its effects on the body. By utilizing mathematical models, such as exponential decay, we can visualize and calculate changes in caffeine concentration over time.
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