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Chapter 2: Problem 6

The temperature \(C\) of a fresh cup of coffee \(t\) minutes after it is poured is given by \(C=125 e^{-0.03 t}+75\) degrees Fahrenheit \(.\) a. Make a graph of \(C\) versus \(t\). b. The coffee is cool enough to drink when its temperature is 150 degrees. When will the coffee be cool enough to drink? c. What is the temperature of the coffee in the pot? (Note: We are assuming that the coffee pot is being kept hot and is the same temperature as the cup of coffee when it was poured.) d. What is the temperature in the room where you are drinking the coffee? (Hint: If the coffee is left to cool a long time, it will reach room temperature.)

Short Answer

Expert verified
a) Graph shows decay approaching 75掳F; b) 17.57 min; c) 200掳F; d) 75掳F.

Step by step solution

01

Understand the Given Function

The temperature function given for the coffee is \( C = 125e^{-0.03t} + 75 \). This represents the temperature in degrees Fahrenheit of the coffee \( t \) minutes after it has been poured.
02

Graph the Function

To graph \( C \) versus \( t \), plot the exponential decay function \( C(t) = 125e^{-0.03t} + 75 \). The y-intercept is at \( t = 0 \), where \( C = 200 \) (since \( 125 + 75 = 200 \)). As \( t \) increases, the exponential term \( 125e^{-0.03t} \) approaches zero, making \( C \) approach 75, the asymptote.
03

Determine When Coffee is Cool Enough to Drink

Set \( C = 150 \) in the equation and solve for \( t \):\[ 125e^{-0.03t} + 75 = 150 \]\[ 125e^{-0.03t} = 75 \]Divide both sides by 125:\[ e^{-0.03t} = 0.6 \]Take the natural logarithm of both sides:\[ -0.03t = \ln(0.6) \]Solve for \( t \):\[ t = \frac{\ln(0.6)}{-0.03} \approx 17.57 \]The coffee is cool enough to drink after approximately 17.57 minutes.
04

Calculate Initial Temperature of Coffee

The initial temperature of the coffee in the pot is when \( t = 0 \). Substitute \( t = 0 \) into the temperature function:\[ C = 125e^{-0.03 \times 0} + 75 = 125 \times 1 + 75 = 200 \]The initial temperature of the coffee is 200 degrees Fahrenheit.
05

Determine Room Temperature

The room temperature is the long-term limit of the temperature function as \( t \) approaches infinity. As \( t \to \infty \), \( e^{-0.03t} \to 0 \), so:\[ C = 125 \times 0 + 75 = 75 \]The room temperature is 75 degrees Fahrenheit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Temperature Function
In this exercise, the temperature of a fresh cup of coffee decreases exponentially over time. The given function is \( C = 125e^{-0.03t} + 75 \) which illustrates this decay. When the coffee is initially poured (at \( t = 0 \)), the temperature is at its highest, \( C = 200 \) degrees Fahrenheit. Over time, the temperature drops towards the room temperature which is 75 degrees Fahrenheit.

This function is a classic example of how exponential decay functions work in real life:
  • **Initial Temperature**: At \( t = 0 \), the coffee is 200掳F.
  • **Decay Rate**: The constant \(-0.03\) in the formula represents the decay rate. A higher absolute value of this constant would mean the coffee cools faster.
  • **End Temperature**: Eventually, as time approaches infinity, the exponential term will approach zero, leading the temperature to settle at 75掳F.
Understanding this function helps in comprehending how quickly items might cool down or warm up under certain conditions.
Graphing Exponential Functions
Graphing the function \( C(t) = 125e^{-0.03t} + 75 \) involves translating an exponential decay graph. The focus is on showing how the coffee鈥檚 temperature decreases over time. Here's a step-by-step on setting up and interpreting the graph:
  • **Y-Intercept**: At \( t = 0 \), \( C \) is 200掳F. Hence, the graph starts at the point \((0, 200)\).
  • **Asymptote**: As \( t \) increases, \( 125e^{-0.03t} \) tends to zero. The graph approaches the horizontal line at \( C = 75 \). This line indicates the room temperature, which the coffee will never actually reach but gets very close to.
  • **General Shape**: The graph is a smooth curve that begins at \( 200 \) and decreases towards \( 75 \). This curve visually represents how fast the temperature drops initially, and how the rate of decrease slows over time.
Graphing helps to visually grasp the behavior of the coffee's cooling, demonstrating both the initial steep decline and eventual plateau.
Solving Exponential Equations
To find out when the coffee is cool enough to drink, we solve the exponential equation where the temperature \( C = 150 \). Here鈥檚 the solution process detailed:

Starting from:\[ 125e^{-0.03t} + 75 = 150 \]
First, you subtract 75 from both sides to isolate the exponential part:
\[ 125e^{-0.03t} = 75 \]
Next, divide by 125 to solve for the exponential component:
\[ e^{-0.03t} = 0.6 \]
Taking the natural logarithm of both sides, we get:
\[ \ln(e^{-0.03t}) = \ln(0.6) \]
Which simplifies to:
\[ -0.03t = \ln(0.6) \]
Finally, solve for \( t \) by dividing both sides by \(-0.03\):
\[ t = \frac{\ln(0.6)}{-0.03} \approx 17.57 \]
This means the coffee will be cool enough to drink roughly 17.57 minutes after it was poured. The exponential equation provides significant insights into real-world decay processes, modeling how quantities fall to a desirable level over time.

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Most popular questions from this chapter

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