*Alberta Diploma Question*

After Forest finishes drinking a cup of coffee on his way to work in the morning, the amount of caffeine present in his body over time is measured and shown in the table below.

**Elapsed Time (h)**

**Amount of Caffeine (mg)**

0

150

1

132

2

116

3

102

4

90

These data can be modelled by an exponential function of the form

C=a⋅bt

where t is the elapsed time, in hours, and C  is the amount of caffeine in Forest’s body, in milligrams.

Write the equation of the exponential function that models the amount of caffeine in Forest’s body over time. Round the value of a  to the nearest *whole number* and the value of b  to the nearest *hundredth.* Remember to use t as your variable.

C=

If Forest had decided to drink two cups of coffee on his way to work in the morning, then which value of the regression function above will change?

### Step 1: Formulate the exponential model

The general form of the exponential model is:
\[
C = a \cdot b^t
\]
Where:
- \(C\) is the amount of caffeine (mg),
- \(a\) is the initial amount of caffeine,
- \(b\) is the decay factor,
- \(t\) is the time elapsed (hours).

From the table:
- At \(t = 0\), \(C = 150\). Therefore, \(a = 150\).
- The value of \(b\) can be determined using another point from the table.

---

### Step 2: Solve for \(b\)
Using the point at \(t = 1\), \(C = 132\):
\[
132 = 150 \cdot b^1
\]
\[
b = \frac{132}{150} = 0.88
\]

---

### Step 3: Write the model equation
Substitute \(a = 150\) and \(b = 0.88\) into the equation:
\[
C = 150 \cdot 0.88^t
\]

---

### Step 4: Analyze the impact of drinking two cups of coffee
If Forest drinks **two cups of coffee**, the **initial amount of caffeine (\(a\))** will double because each cup contains 150 mg of caffeine. Thus, the new \(a\) would be \(2 \cdot 150 = 300\). However, the **decay factor (\(b\))** remains the same, as the rate at which caffeine is metabolized does not depend on the initial quantity.

---

### Final Answer:
1. The equation of the exponential function is:
   \[
   C = 150 \cdot 0.88^t
   \]

2. If Forest drinks two cups of coffee, the **initial amount of caffeine (\(a\))** will change, but the decay factor (\(b\)) will remain the same.

---

### Additional Questions:
1. How does the half-life of caffeine relate to the decay factor \(b\)?
2. Can you find the half-life of caffeine based on the given data?
3. What happens to the graph of the function if Forest drinks three cups of coffee instead of one?
4. How would you interpret \(b = 0.88\) in terms of caffeine metabolism?
5. What other real-world processes can be modeled using an exponential decay function?

---

### Tip:
To better understand exponential decay, compare the caffeine amounts after each hour and see how they form a consistent percentage decrease. This is the hallmark of exponential decay.

After Forest finishes drinking a cup of coffee on his way to work in the morning, the amount of caffeine present in his body over time is measured and shown in the table below. Write the equation of the exponential function that models the amount of caffeine in Forest’s body over time. Round the value of a to the nearest whole number and the value of b to the nearest hundredth. Remember to use t as your variable. If Forest had decided to drink two cups of coffee on his way to work in the morning, then which value of the regression function above will change?

Discover how to model the decay of caffeine in the body using exponential functions. This problem explores the equation C = a * b^t, where the initial amount of caffeine and decay factor are determined from data. Learn how doubling the caffeine intake affects the model. Perfect for Grades 10-12 students learning about exponential decay and regression.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation