Problem 4
Question
The temperature, \(H\), in degrees Celsius, of a cup of coffee placed on the kitchen counter is given by \(H=f(t)\) where \(t\) is in minutes since the coffee was put on the counter. (a) Is \(f^{\prime}(t)\) positive or negative? Give a reason for your answer. (b) What are the units of \(f^{\prime}(20) ?\) What is its practical meaning in terms of the temperature of the coffee?
Step-by-Step Solution
Verified Answer
(a) \( f'(t) \) is negative because the temperature decreases. (b) The units are °C/min, indicating the cooling rate.
1Step 1: Understand the Problem
The function \( f(t) \) represents the temperature of coffee over time. We need to determine whether the derivative \( f'(t) \) is positive or negative, and interpret its meaning and units at \( t = 20 \).
2Step 2: Analyze the Behavior of Temperature Over Time
Typically, when a hot object like coffee is exposed to room temperature, its temperature decreases over time. Thus, \( f(t) \) will decrease as \( t \) increases.
3Step 3: Determine the Sign of \( f'(t) \)
The derivative \( f'(t) \) represents the rate of change of temperature with respect to time. Since the temperature decreases over time, \( f'(t) \) is negative.
4Step 4: Determine the Units of \( f'(20) \)
The derivative \( f'(t) \) represents the rate of temperature change per unit time. If \( H \) is in degrees Celsius and \( t \) is in minutes, the units of \( f'(20) \) are degrees Celsius per minute.
5Step 5: Interpret \( f'(20) \) in Practical Terms
The value \( f'(20) \) gives the rate at which the coffee is cooling down 20 minutes after being placed on the counter. A negative value indicates the temperature is decreasing.
Key Concepts
Temperature FunctionsDerivativesUnits of Measurement
Temperature Functions
Temperature functions play a crucial role when studying the behavior of substances as they gain or lose heat over time. In our example, we have a cup of coffee, whose temperature is represented by a function \( H = f(t) \). Here, \( H \) denotes the coffee's temperature in degrees Celsius, and \( t \) is time in minutes since the coffee is left on the counter. By modeling the temperature with a function, we can predict how the coffee will cool down as time progresses.
Understanding how temperature functions behave helps in determining heat transfer rates and predicting when an item will reach room temperature. Typically, a hot object will lose heat to its surroundings, which in this case means the coffee temperature decreases over time. Thus, \( f(t) \) decreases continuously.
Because of this trend, if we're asked about the rate of change of temperature over time, we naturally think of derivatives, as they can tell us how fast or slow these changes occur.
Understanding how temperature functions behave helps in determining heat transfer rates and predicting when an item will reach room temperature. Typically, a hot object will lose heat to its surroundings, which in this case means the coffee temperature decreases over time. Thus, \( f(t) \) decreases continuously.
Because of this trend, if we're asked about the rate of change of temperature over time, we naturally think of derivatives, as they can tell us how fast or slow these changes occur.
Derivatives
Derivatives are fundamental in calculus for understanding how a quantity changes with respect to another. In temperature functions, the derivative \( f'(t) \) specifically tells us the rate of change of temperature with time.
In our problem, this is expressed as how rapidly the coffee's temperature changes with each passing minute. Since the coffee cools over time, \( f'(t) \) is negative, indicating a decrease in temperature. When we say \( f'(t) \) is negative, it helps us know that the coffee is losing heat. The larger the magnitude of the negative value, the quicker the cooling process.
In our problem, this is expressed as how rapidly the coffee's temperature changes with each passing minute. Since the coffee cools over time, \( f'(t) \) is negative, indicating a decrease in temperature. When we say \( f'(t) \) is negative, it helps us know that the coffee is losing heat. The larger the magnitude of the negative value, the quicker the cooling process.
- \( f'(t) > 0 \): Temperature is increasing (not our case with cooling coffee)
- \( f'(t) = 0 \): Temperature remains constant
- \( f'(t) < 0 \): Temperature is decreasing
Units of Measurement
Units of measurement give context to mathematical expressions, making them practical and applicable to real-life situations. For the derivative \( f'(t) \) of our temperature function, the units are derived from the units of \( H \) and \( t \).
Since \( H \) is measured in degrees Celsius and \( t \) is in minutes, \( f'(t) \), which is the rate of change of temperature with respect to time, has units of degrees Celsius per minute. It essentially answers the question: how many degrees does the coffee cool down per minute?
Since \( H \) is measured in degrees Celsius and \( t \) is in minutes, \( f'(t) \), which is the rate of change of temperature with respect to time, has units of degrees Celsius per minute. It essentially answers the question: how many degrees does the coffee cool down per minute?
- Degrees Celsius: Measures temperature
- Minutes: Measures time
- Degrees Celsius per minute: Rate of temperature change
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