Question

The temperature \(F\) in degrees Fahrenheit is related to the temperature \(C\) in degrees Celsius by the equation \(F=\frac{9}{5} C+32\) a. Find an equation relating \(d F / d t\) to \(d C / d t\) b. How fast is the temperature in an oven changing in degrees Fahrenheit per minute if it is rising at \(10^{\circ}\) Celsius per min?

Short answer

Answer: 18°F per minute.

Step-by-step solution

Differentiate the given equation with respect to time

We know that \(F=\frac{9}{5} C+32\). To find the relationship between the rates of change, we will implicitly differentiate both sides of this equation with respect to time \(t\). So we will have: $$ \frac{dF}{dt} = \frac{9}{5} \frac{dC}{dt}$$ **Phase 2: Find the rate of change in degrees Fahrenheit**

Plug in the given rate of change in degrees Celsius

We are given that the temperature in Celsius is rising at a rate of \(10^{\circ}\)C per minute, which means \(\frac{dC}{dt} = 10\). Now let's plug this value into the equation we found in Phase 1: $$ \frac{dF}{dt} = \frac{9}{5} \cdot 10$$

Calculate the rate of change in degrees Fahrenheit

Now, we simply calculate the right-hand side of the equation: $$ \frac{dF}{dt} = \frac{9}{5} \cdot 10 = 18$$ So, the temperature in an oven is increasing at a rate of 18 degrees Fahrenheit per minute.

Key concepts

Understanding Related Rates

Related rates problems are a fascinating application of differential calculus, which typically involve two or more quantities that change over time. These quantities are related by an equation, and the goal is to determine how the rate of change of one variable affects the rate of change of another.

For example, consider our problem at hand where the temperature in Fahrenheit and Celsius are related by the equation \( F=\frac{9}{5}C+32 \). Suppose we're interested in finding out how quickly the temperature is changing in Fahrenheit when we already know the rate at which it is changing in Celsius. We would use the concept of related rates to establish a connection between \( \frac{dF}{dt} \) and \( \frac{dC}{dt} \).

It's crucial to recognize that implicitly differentiating the equation that relates the two variables with respect to time will yield the desired relationship between their rates. The understanding of how different rates of change influence each other is not only pertinent in temperature changes but also in various real-world scenarios like motion, economics, and fluid dynamics, to name a few.

Differential Calculus in Action

Differential calculus is a branch of mathematics that studies how things change, with the derivative being its fundamental tool. A derivative measures how a function changes as its input changes — in simpler terms, it's the rate of change. That's why it's a perfect match for solving related rates problems.

Implicit differentiation, as used in our temperature example, is a technique where we find the derivative of a function with respect to a variable, even though the function may not be solved for that variable. When differentiating, we apply the chain rule which allows us to find derivatives of composite functions.

To illustrate, in our scenario, we differentiated the temperature conversion formula with respect to time to find how fast the Fahrenheit temperature changes as the Celsius temperature changes. This process demonstrates the power of differential calculus to solve real-life problems involving rates of change in a dynamic way.

Temperature Conversion and Rates

Temperature conversion is a necessary process when comparing readings in different temperature scales, such as Celsius and Fahrenheit. The relationship between these scales is given by the linear equation \( F=\frac{9}{5}C+32 \), where \(F\) represents degrees Fahrenheit, and \(C\) represents degrees Celsius.

In our problem, the rate of change for temperature in Celsius, denoted as \( \frac{dC}{dt} \), was provided, and the goal was to find out the rate of change in Fahrenheit, \( \frac{dF}{dt} \). Using our understanding of the temperature conversion formula and the principles of related rates, we determined that the Fahrenheit temperature changes at a rate of \(18^\circ F\) per minute when the Celsius temperature changes at a rate of \(10^\circ C\) per minute.

This type of calculation is not only academic but has practical applications in various fields, such as climatology, cooking, and many other scenarios where understanding the rate of temperature change is critical.