https://file.aiquickdraw.com/mathbot/file/1728528681328af4rcthz.jpg

Let's analyze and solve the problems shown in the image.

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**Problem 7:**
The function \( f(t) = 32.744e^{-0.252t} \) (where \( t = 1 \) corresponds to the 2011 season) represents the average number of viewers (in millions) for Fox's *American Idol* from the 2011 to 2013 seasons.

(a) **What was the average number of viewers in the 2011 season?**

For \( t = 1 \), the function becomes:
\[
f(1) = 32.744e^{-0.252 \times 1} = 32.744e^{-0.252}
\]

Calculating \( e^{-0.252} \), we find:
\[
f(1) = 32.744 \times 0.777 = 25.429 \text{ million viewers}
\]

This matches the incorrect answer marked in the image as **25.420 million**.

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(b) **What was the average number of viewers in the 2014 season?**

Since the trend continued, we calculate for \( t = 4 \) (as 2014 is three years after 2011):
\[
f(4) = 32.744e^{-0.252 \times 4} = 32.744e^{-1.008}
\]
\[
f(4) = 32.744 \times 0.365 = 11.950 \text{ million viewers}
\]

This is correct and matches the given solution of **11.950 million**.

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**Problem 8:**
The concentration of a drug in an organ is given by the function:
\[
x(t) = 0.06 + 0.16(1 - e^{-0.02t})
\]
where \( x(t) \) is measured in mg/cm³ and \( t \) is in seconds.

(a) **What is the initial concentration of the drug in the organ?**

For \( t = 0 \):
\[
x(0) = 0.06 + 0.16(1 - e^{-0.02 \times 0})
\]
Since \( e^0 = 1 \), the equation simplifies to:
\[
x(0) = 0.06 + 0.16(1 - 1) = 0.06 \, \text{mg/cm}^3
\]
Thus, the initial concentration is **0.06 mg/cm³**.

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(b) **What is the concentration of the drug in the organ after 20 seconds?**

For \( t = 20 \):
\[
x(20) = 0.06 + 0.16(1 - e^{-0.02 \times 20})
\]
Simplifying:
\[
x(20) = 0.06 + 0.16(1 - e^{-0.4}) = 0.06 + 0.16(1 - 0.6703)
\]
\[
x(20) = 0.06 + 0.16(0.3297) = 0.06 + 0.0528 = 0.1128 \, \text{mg/cm}^3
\]
So, the concentration after 20 seconds is **0.1128 mg/cm³**.

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Let me know if you'd like any additional details or explanations! Here are five related questions to explore further:

1. How does the decay constant in the function \( f(t) \) affect the rate of viewer decline?
2. What would the average number of viewers be in 2015 if the trend continued?
3. How would you calculate the time it takes for the drug concentration to reach half its initial value?
4. Could the exponential decay model for viewers apply to other TV shows? How would you adapt it?
5. How can you determine the maximum concentration of the drug in the organ?

**Tip:** When dealing with exponential models, always check the units of time and rates to ensure proper interpretation of decay rates or growth factors.

The concentration of a drug in an organ is given by the function x(t) = 0.06 + 0.16(1 - e^{-0.02t}). (a) What is the initial concentration of the drug in the organ? (b) What is the concentration of the drug in the organ after 20 seconds?

Solve an exponential decay problem related to drug concentration. This problem covers the function x(t) = 0.06 + 0.16(1 - e^{-0.02t}) and asks for the initial concentration and the concentration after 20 seconds. It includes calculations for both parts using exponential decay properties.

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