Math Problem Statement
A thermometer reading 8degreesC is brought into a room with a constant temperature of 30 degrees Upper C. (a) If the thermometer reads 13degreesC after 3 minutes, what will it read after being in the room for 5 minutes? For 10 minutes? (b) Graph the relation between temperature and time. Use TRACE to verify that your answers are correct.
Solution
To solve this problem, we'll use Newton's Law of Cooling, which states that the rate of change of temperature of an object is proportional to the difference between its current temperature and the ambient temperature. The law is mathematically represented as:
Where:
- is the temperature of the object at time .
- is the ambient (room) temperature.
- is a positive constant that depends on the properties of the object.
- is the time in minutes.
Given:
- Initial temperature,
- Room temperature,
- Temperature after 3 minutes,
Let's solve step by step:
Step 1: Find the general solution for
Newton's Law of Cooling can be integrated to give the formula:
Where:
- is the initial temperature of the thermometer (8°C in this case).
So, the equation becomes:
Simplifying:
Step 2: Find using the given data
We know that :
Rearranging this equation:
Thus:
Taking the natural logarithm of both sides:
Calculating this value:
Step 3: Find the temperature after 5 and 10 minutes
Now we can plug the values of and into the equation .
Temperature after 5 minutes:
Calculating:
Temperature after 10 minutes:
Calculating:
Step 4: Graph the relationship between temperature and time
The graph of will show an exponential curve starting at and approaching as increases.
To verify these calculations using a graphing tool, plot the function and use TRACE to check the temperatures at and .
Would you like a detailed graph plotted, or any further explanation?
5 Related Questions:
- How would the result change if the ambient temperature were different?
- What happens to the rate of temperature change as the thermometer's temperature approaches room temperature?
- Can you derive the formula for Newton's Law of Cooling from first principles?
- How does the constant affect the speed of cooling or heating?
- What are some practical applications of Newton's Law of Cooling?
Tip:
Newton's Law of Cooling can also be applied in reverse for heating problems where the object is initially cooler than its surroundings.
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Math Problem Analysis
Mathematical Concepts
Calculus
Exponential Decay
Newton's Law of Cooling
Formulas
Newton's Law of Cooling: dT/dt = -k(T - T_ambient)
Temperature Equation: T(t) = T_ambient + (T_0 - T_ambient) * e^(-kt)
Theorems
Newton's Law of Cooling
Suitable Grade Level
Grades 11-12 (or introductory college level)
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