If a cup of coffee has temperature 93°C in a room where the ambient air temperature is 20°C, then, according to Newton's Law of Cooling, the temperature of the coffee after t minutes is T(t) = 20 +73e-t/55 What is the average temperature of the coffee during the first 26 minutes? average temp = . °C
If a cup of coffee has temperature 93°C in a room where the ambient air temperature is 20°C, then, according to Newton's Law of Cooling, the temperature of the coffee after t minutes is T(t) = 20 +73e-t/55 What is the average temperature of the coffee during the first 26 minutes? average temp = . °C
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Newton's Law of Cooling: Average Temperature Calculation**
**Problem Statement:**
If a cup of coffee has a temperature of 93°C in a room where the ambient air temperature is 20°C, then, according to Newton's Law of Cooling, the temperature of the coffee after \( t \) minutes is given by:
\[ T(t) = 20 + 73e^{-\frac{t}{55}} \]
**Question:**
What is the average temperature of the coffee during the first 26 minutes?
**Solution:**
To find the average temperature, you will need to calculate the definite integral of the temperature function \( T(t) \) over the interval from \( t = 0 \) to \( t = 26 \) and then divide by the length of the interval (26 minutes).
**Formula:**
\[ \text{Average Temperature} = \frac{1}{26} \int_{0}^{26} \left( 20 + 73e^{-\frac{t}{55}} \right) dt \]
Please input the average temperature calculation in the box provided:
\[ \text{average temp} = \_\_\_\_ \text{°C} \]
**Graphical Representation:**
While not provided here, a possible graph to accompany this explanation would illustrate the exponential decay curve of the coffee's temperature over time, starting at 93°C and approaching 20°C as time increases. This visualization helps to understand how the temperature changes dynamically.
To calculate the average, you would use integral calculus techniques, which provide a precise way to determine the mean value of such non-linear functions over specified intervals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9659bf0-1b68-44e7-9939-e8c666150000%2Fe819c3bf-c466-4a9f-ab20-afd26ce12fa4%2Ff95urp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Newton's Law of Cooling: Average Temperature Calculation**
**Problem Statement:**
If a cup of coffee has a temperature of 93°C in a room where the ambient air temperature is 20°C, then, according to Newton's Law of Cooling, the temperature of the coffee after \( t \) minutes is given by:
\[ T(t) = 20 + 73e^{-\frac{t}{55}} \]
**Question:**
What is the average temperature of the coffee during the first 26 minutes?
**Solution:**
To find the average temperature, you will need to calculate the definite integral of the temperature function \( T(t) \) over the interval from \( t = 0 \) to \( t = 26 \) and then divide by the length of the interval (26 minutes).
**Formula:**
\[ \text{Average Temperature} = \frac{1}{26} \int_{0}^{26} \left( 20 + 73e^{-\frac{t}{55}} \right) dt \]
Please input the average temperature calculation in the box provided:
\[ \text{average temp} = \_\_\_\_ \text{°C} \]
**Graphical Representation:**
While not provided here, a possible graph to accompany this explanation would illustrate the exponential decay curve of the coffee's temperature over time, starting at 93°C and approaching 20°C as time increases. This visualization helps to understand how the temperature changes dynamically.
To calculate the average, you would use integral calculus techniques, which provide a precise way to determine the mean value of such non-linear functions over specified intervals.
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