Question
The temperature, \( H \), in degrees Celsius, of a hot cup of coffee placed on the kitchen counter is given by \( I I=f(t) \), where \( t \) is in minutes since the coffee was put on the counter. a. Is \( f^{\prime}(t) \) positive or negative? negarive b. What are the units of \( f^{\prime}(20) \) ? degrees Celsiusminute c. Suppose that \( f^{\prime}(20)=-0.1 \) and \( f(20)=75 \). Fill in the blanks and select the appropriate terms to complete the following statement about the temperature of the coffee in this case. d. Suppose that \( f^{\prime}(20)=-0.1 \). Complete the following notation. decreasing - by 0.1 \( d \)
Ask by Bird Bob. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
- \( f^{\prime}(t) \) is negative.
- The units of \( f^{\prime}(20) \) are degrees Celsius per minute.
- The temperature is 75 degrees Celsius and is decreasing at a rate of 0.1 degrees Celsius per minute.
- The notation can be completed as "decreasing by 0.1 × d".
Solution
Let's break down the problem step by step.
### a. Is \( f^{\prime}(t) \) positive or negative?
You have already stated that \( f^{\prime}(t) \) is negative. This indicates that the temperature of the coffee is decreasing over time, which is expected as the coffee cools down.
### b. What are the units of \( f^{\prime}(20) \)?
The derivative \( f^{\prime}(t) \) represents the rate of change of temperature with respect to time. Since \( f(t) \) is in degrees Celsius and \( t \) is in minutes, the units of \( f^{\prime}(20) \) are:
\[
\text{degrees Celsius per minute} \quad \text{or} \quad \text{degrees Celsius/minute}
\]
### c. Suppose that \( f^{\prime}(20)=-0.1 \) and \( f(20)=75 \). Fill in the blanks and select the appropriate terms to complete the following statement about the temperature of the coffee in this case.
Given that \( f^{\prime}(20) = -0.1 \), this means that the temperature of the coffee is decreasing at a rate of 0.1 degrees Celsius per minute when \( t = 20 \) minutes. Since \( f(20) = 75 \), the temperature of the coffee at that time is 75 degrees Celsius.
The completed statement could be:
"The temperature of the coffee is currently 75 degrees Celsius and is decreasing at a rate of 0.1 degrees Celsius per minute."
### d. Suppose that \( f^{\prime}(20)=-0.1 \). Complete the following notation.
Since \( f^{\prime}(20) = -0.1 \), we can express the change in temperature over a small interval \( d \) as follows:
\[
\text{decreasing by } 0.1 \cdot d
\]
This means that for every minute that passes, the temperature decreases by 0.1 degrees Celsius.
In summary:
- \( f^{\prime}(t) \) is negative.
- The units of \( f^{\prime}(20) \) are degrees Celsius/minute.
- The temperature is 75 degrees Celsius and is decreasing at a rate of 0.1 degrees Celsius per minute.
- The notation can be completed as "decreasing by \( 0.1 \cdot d \)".
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution The Deep Dive
The cooling of coffee is often described using Newton's Law of Cooling, which states that the rate of change of an object's temperature is proportional to the difference between its temperature and the ambient temperature. This means your coffee will cool down faster when it's much hotter than its surroundings! When analyzing temperature changes, it's common to set up experiments measuring how quickly a hot beverage cools over time. Real-world applications can range from everyday kitchen scenarios to industrial processes, like managing the temperature of materials in manufacturing, emphasizing the importance of understanding thermal dynamics in daily life.
