Problem 366 - Calculus Volume 1

Question

For the following exercises, use \(y=y_{0} e^{k t}\). You are cooling a turkey that was taken out of the oven with an internal temperature of \(165^{\circ} \mathrm{F}\). After 10 minutes of resting the turkey in a \(70^{\circ} \mathrm{F}\) apartment, the temperature has reached \(155^{\circ} \mathrm{F}\). What is the temperature of the turkey 20 minutes after taking it out of the oven?

Step-by-Step Solution

Answer

The temperature of the turkey after 20 minutes is approximately 145.3°F.

Step 1: Identify the Known Values

We need to identify the initial conditions of the problem. Initially, the turkey's temperature is \(y_0 = 165^{\circ} \mathrm{F}\). Ambient temperature is given as \(70^{\circ} \mathrm{F}\). After resting 10 minutes, the temperature drops to \(155^{\circ} \mathrm{F}\). We need to calculate the temperature after 20 minutes.

Step 2: Determine the Exponential Decay Model

The formula for exponential cooling is \(y = y_0 e^{kt}\). We can express this in terms of time \(t = 0\) as \(y_0 = 165\) when removing from the oven and \(y = 155\) when \(t = 10\). Substitute these into the equation to solve for \(k\).

Step 3: Solve for the Rate Constant \(k\)

Substitute \(y = 155\), \(y_0 = 165\), and \(t = 10\) into the formula: \[155 = 165 \, e^{10k}\]Divide both sides by 165:\[\frac{155}{165} = e^{10k}\]Take the natural logarithm of both sides to solve for \(k\): \[k = \frac{1}{10} \ln\left(\frac{155}{165}\right)\]Calculate this value.

Step 4: Apply the Exponential Decay Function for 20 Minutes

Using the value of \(k\) calculated in Step 3, we substitute \(t = 20\) in \(y = y_0 e^{kt} \) to find the temperature of the turkey after 20 minutes:\[y = 165 \, e^{20k}\]Simplify and calculate the result to find the turkey's temperature at \(t = 20\).

Step 5: Calculate Final Temperature

We use the value of \(k\) calculated earlier in the formula from Step 4: Substitute back and solve to get:\[y = 165 \, e^{20 \times \frac{1}{10} \ln\left(\frac{155}{165}\right)}\]Simplifying, calculate \(e^{20k}\) which gives us the temperature of the turkey after 20 minutes.

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. These functions are often represented as \( y = y_0 e^{kt} \), where:

• \( y_0 \) is the initial value.
• \( k \) is the rate constant.
• \( t \) is time.
• \( e \) is the base of the natural logarithm, approximately equal to 2.718.

Exponential functions are crucial in modeling situations of exponential decay or growth.
In this type of functions, the base remains consistent while the exponent changes.
Depending on the sign of \( k \), the function can represent decay or growth:

• If \( k \) is negative, it describes decay.
• If \( k \) is positive, it shows growth.

These characteristics make exponential functions essential for understanding different real-world phenomena, like population growth or radioactive decay.

Cooling Process

The cooling process discussed here refers to how an object's temperature decreases over time, specifically following an exponential decay model. This process can be illustrated by Newton's Law of Cooling, which is expressed mathematically as \( y = y_0 e^{kt} \).

• \( y_0 \) is the initial temperature of the object.
• \( y \) is the temperature of the object at time \( t \).
• \( k \) is a negative rate constant reflecting the cooling rate.

The example of cooling a turkey involves finding its temperature at a specific time after it's removed from the oven. Initially, the turkey is \(165^\circ \mathrm{F}\) and cools at a rate dependent on the ambient temperature, \(70^\circ \mathrm{F}\).
The process is termed "cooling" when temperatures decrease over time, modeling an exponential decay. The rate constant \( k \) is crucial for determining how quickly this cooling process occurs.
In this context, the cooling rate is affected by external factors like ambient temperature, which influences the value of \( k \).

Natural Logarithm

The natural logarithm, abbreviated as "ln," is a logarithm with a base of \( e \). It plays a pivotal role in solving exponential equations, especially when determining the rate constant \( k \) in exponential decay models.
To solve for \( k \) in the exponential equation \( y = y_0 e^{kt} \), we often need to apply the natural logarithm to both sides of the equation:

• Take \( ln \) to linearize the equation, making \( kt \) a straightforward multiplier.
• For any number \( x \), \( ln(e^x) = x \) simplifies calculations.

For instance, to calculate \( k \) in our turkey-cooling problem, we rearrange the equation to isolate \( e^{10k} \), then apply the natural logarithm:

• The equation becomes \( k = \frac{1}{10} \ln\left(\frac{155}{165}\right) \).
• This step turns the exponential relationship into a linear one, making \( k \) easier to solve.

Natural logarithms smoothly convert complex exponential equations into manageable arithmetic, facilitating the analysis of situations like the turkey cooling process.