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Chapter 4: Problem 81

Cooking a Roast James knows that to get well-done beef, it should be brought to a temperature of \(170^{\circ} \mathrm{F}\). He placed a sirloin tip roast with a temperature of \(35^{\circ} \mathrm{F}\) in an oven with a temperature of \(325^{\circ},\) and after 3 hr the temperature of the roast was \(140^{\circ} .\) How much longer must the roast be in the oven to get it well done? If the oven temperature is set at \(170^{\circ},\) how long will it take to get the roast well done? HINT The difference between the roast temperature and the oven temperature decreases exponentially.

Short Answer

Expert verified
Solve for the cooling constant \( k \), then determine the additional time for the roast to reach well done. Finally, calculate the total time needed if the oven were set to \( 170^{\circ} \mathrm{F}\).

Step by step solution

01

- Understand the problem

The goal is to determine the additional time needed to bring the roast from an internal temperature of \(140^{\circ} \mathrm{F}\) to \(170^{\circ} \mathrm{F}\) in a \(325^{\circ} \mathrm{F}\) oven. Additionally, calculate the total time to cook the roast from \(35^{\circ} \mathrm{F}\) to \(170^{\circ} \mathrm{F}\) in a \(170^{\circ} \mathrm{F}\) oven based on the given exponential cooling formula.
02

- Use Newton's Law of Cooling

According to Newton's Law of Cooling, the temperature of the roast can be expressed as: \( T(t) = T_{\text{env}} + (T_0 - T_{\text{env}}) e^{-kt} \). Where \(T_{\text{env}}\) is the oven temperature, \(T_0\) is the initial temperature of the roast, \(T(t)\) is the temperature of the roast after time \(t\), and \(k\) is the cooling constant.
03

- Apply given data to compute \(k\)

Set \(T_{\text{env}} = 325\), \(T_0 = 35\). After 3 hours, the temperature of the roast is \(140^{\circ} \mathrm{F}\). Substitute these values into the equation: \ \ \[ 140 = 325 + (35 - 325) e^{-3k} \] \ Solve for \(k\).
04

- Simplify equation and solve for \(k\)

Subtract 325 from both sides: \ \[ 140 - 325 = (35 - 325) e^{-3k} \] \ \[ -185 = -290 e^{-3k} \] \ Divide by -290: \ \[ \frac{185}{290} = e^{-3k} \] \ Apply the natural logarithm to both sides: \ \[ \ln (\frac{185}{290}) = -3k \] \ \[ k = -\frac{1}{3} \ln (\frac{185}{290}) \]
05

- Calculate additional time to \(170^{\circ} \mathrm{F}\)

Using \(k\), find the time needed to reach \(170^{\circ} \mathrm{F}\). Substitute \(T(3 + t') = 170\) and \(T_{\text{env}} = 325\), then solve: \ \ \[ 170 = 325 + (140 - 325) e^{-kt'} \] \ Solving for \(t'\) will give additional time needed.
06

- Set oven temperature to \(170^{\circ} \mathrm{F}\)

When the oven is set to \(170^{\circ} \mathrm{F}\), we need the entire cooking time using the initial temperature \(35^{\circ} \mathrm{F}\) and environmental temperature \(170^{\circ} \mathrm{F}\). Use the equation: \ \ \[ T(t) = 170 + (35 - 170) e^{-kt} \] \ and solve for \(t\) when \(T(t) = 170\).

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Key Concepts

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

Exponential Decay
Exponential decay is a process where the quantity decreases at a rate proportional to its current value. In the context of Newton's Law of Cooling, we use exponential decay to model the rate at which the temperature difference between an object and its environment decreases over time. The general formula for exponential decay can be expressed as: \[ N(t) = N_0 e^{-kt} \] Here, \( N(t) \) is the quantity at time \( t \), \( N_0 \) is the initial quantity, \( k \) is the decay constant, and \( e \) is the base of the natural logarithm. This formula resembles the cooling equation used in the problem with temperatures.
Temperature Modeling
Temperature modeling involves predicting how the temperature of an object changes over time within a certain environment. We use mathematical formulas based on initial conditions and environmental factors to create these models. In our example, we're looking at how a roast's temperature changes in an oven using Newton's Law of Cooling. The equation \( T(t) = T_{env} + (T_0 - T_{env}) e^{-kt} \) helps predict the temperature \( T(t) \) after time \( t \). Here, \( T_{env} \) is the stable oven temperature, \( T_0 \) is the roast's initial temperature, and \( k \) is a constant related to how quickly the temperature changes.
Differential Equations
Differential equations are equations that involve the derivatives of a function. They show how a quantity changes over time, and they're used in many fields, like physics, engineering, and finance. Newton's Law of Cooling itself is derived from a first-order differential equation: \[ \frac{dT}{dt} = -k (T - T_{env}) \] This equation says that the rate of change of the temperature \( \frac{dT}{dt} \) is proportional to the difference between the object's temperature \( T \) and the environmental temperature \( T_{env} \). Solving this differential equation gives us the exponential decay formula used in the roast problem.

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