Question

Thawing a Package of Steaks You take a three-pound package of steaks out of the freezer at 11 A.M. and place it in the refrigerator. Will the steaks be thawed in time to be grilled at 6 p.m.? Assume that the refrigerator temperature is \(40^{\circ} \mathrm{F}\) and that the freezer temperature is \(0^{\circ} \mathrm{F}\). Use the formula for Newton's Law of Cooling \(t=-5.05 \ln \frac{T-40}{0-40}\) where \(t\) is the time in hours (with \(t=0\) corresponding to 11 A.M.) and \(T\) is the temperature of the package of steaks (in degrees Fahrenheit).

Short answer

Depend upon the value calculated for \(T\) (in Fahrenheit). If \(T\) is equal to or close to 40 (∼room temperature), it implies that the steak will have thawed by 6 p.m. If \(T\) is significantly less than 40, the steak will not be completely thawed by that time. The exact numerical value will depend upon calculator computations.

Step-by-step solution

Identify Knowns and Unknowns

From the problem, we know that the steak is taken out of the freezer at 11 a.m. (\(t=0\)) and has an initial temperature of \(0^{\circ} \mathrm{F}\). We also know that the temperature of the refrigerator is \(40^{\circ}F\). We want to know whether the steak has reached the refrigerator's temperature by 6 p.m. (\(t=7\)), thus the temperature \(T\) after 7 hours is unknown.

Substitute Values into the Equation

Substitute \(t=7\) hours and the initial and final temperatures into Newton's law of cooling. We get \(7=-5.05 \ln \frac{T-40}{0-40}\).

Solve for T

First, we need to invert the sign on the right side of the equation and simplify. We get \(-7=5.05 \ln \frac{T-40}{-40}\). Rewriting the logarithm as an exponent gives us \(\e^{-7/5.05}=\frac{T-40}{-40}\). Solve for \(T\) by multiplying both sides with -40 and adding 40.

Calculate T

Perform the operations using a calculator to find the value of \(T\), which should be the temperature of the steak at 6 p.m.

Key concepts

Exponential and Logarithmic Equations

When working with Newton's Law of Cooling, we frequently encounter exponential and logarithmic equations. These are a special category of equations where the variable, or the unknown we're trying to solve for, is located in the exponent or within a logarithm. In our example with the steaks, the given formula is logarithmic, and reads as follows:

\[ t = -5.05 \ln\left(\frac{T-40}{0-40}\right) \]
To solve for the steak's temperature, we made use of the properties of logarithms, which allow us to convert logarithmic statements into exponential form and vice versa. This flexibility is key in simplifying the steps needed to find our temperature, \(T\). Understanding how to manipulate these equations can open up a way to solve many problems in physics, chemistry, and other fields where rates of change are modeled with similar mathematical expressions.

Temperature Conversion

Temperature conversion can often come into play when we're dealing with physical sciences, as different measurements may require different temperature scales like Celsius, Fahrenheit, or Kelvin. In the given problem, there is no immediate need for temperature conversion as both the freezer and refrigerator temperatures are already in Fahrenheit. However, when working with scientific data or international standards, you might be expected to convert temperatures to or from Fahrenheit. For example, to convert Fahrenheit to Celsius, you would use the formula

\[ C = \frac{5}{9}(F - 32) \]
It's important to be familiar with these conversions because the rate of cooling can vary if temperatures are represented in different scales, which may influence the calculations in thermodynamics or when using formulas like Newton's Law of Cooling.

Applied Mathematics

The study of Newton's Law of Cooling is an excellent illustration of how mathematics can be applied to real-world phenomena, such as predicting how long it will take for food to thaw. In applied mathematics, we use mathematical models to describe and solve concrete problems.

To resolve our question regarding the steak thawing process, we applied Newton's formula, which is an equation derived from empirical observations and thermodynamics first conceptualized by Sir Isaac Newton. By using this formula, we attempt to calculate a specific value that can guide our decision-making process. This is what makes applied mathematics incredibly valuable; it provides the tools to translate theoretical understanding into practical applications that have tangible implications in everyday life, such as ensuring your steaks are ready for the grill on time.

Rate of Cooling

The rate of cooling describes how fast an object is losing heat over time. According to Newton's Law of Cooling, the rate at which an object cools down is proportional to the difference between the temperature of the object and that of its surroundings.

This relationship is elegantly expressed in the logarithmic equation we've been working with. It tells us just how quickly the steak package approaches the fridge's stable temperature. The '-5.05' in the equation is a constant that relates to this rate for our specific situation. The more significant the temperature difference between the object and its environment, the faster it should cool, at least until the temperatures begin to equalize. Knowledge of cooling rates is particularly useful in a variety of applications including food safety, material sciences, and even forensic investigations to determine the time of death.