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:

dTdt=k(TTambient)\frac{dT}{dt} = -k (T - T_{\text{ambient}})

Where:

  • TT is the temperature of the object at time tt.
  • TambientT_{\text{ambient}} is the ambient (room) temperature.
  • kk is a positive constant that depends on the properties of the object.
  • tt is the time in minutes.

Given:

  • Initial temperature, T(0)=8CT(0) = 8^\circ C
  • Room temperature, Tambient=30CT_{\text{ambient}} = 30^\circ C
  • Temperature after 3 minutes, T(3)=13CT(3) = 13^\circ C

Let's solve step by step:

Step 1: Find the general solution for T(t)T(t)

Newton's Law of Cooling can be integrated to give the formula:

T(t)=Tambient+(T0Tambient)ektT(t) = T_{\text{ambient}} + (T_0 - T_{\text{ambient}}) e^{-kt}

Where:

  • T0T_0 is the initial temperature of the thermometer (8°C in this case).

So, the equation becomes:

T(t)=30+(830)ektT(t) = 30 + (8 - 30) e^{-kt}

Simplifying:

T(t)=3022ektT(t) = 30 - 22 e^{-kt}

Step 2: Find kk using the given data

We know that T(3)=13CT(3) = 13^\circ C:

13=3022e3k13 = 30 - 22 e^{-3k}

Rearranging this equation:

22e3k=3013=1722 e^{-3k} = 30 - 13 = 17

Thus:

e3k=1722e^{-3k} = \frac{17}{22}

Taking the natural logarithm of both sides:

3k=ln(1722)-3k = \ln \left(\frac{17}{22}\right)

k=13ln(1722)k = -\frac{1}{3} \ln \left(\frac{17}{22}\right)

Calculating this value:

k0.088k \approx 0.088

Step 3: Find the temperature after 5 and 10 minutes

Now we can plug the values of kk and tt into the equation T(t)=3022e0.088tT(t) = 30 - 22 e^{-0.088t}.

Temperature after 5 minutes:

T(5)=3022e0.088×5T(5) = 30 - 22 e^{-0.088 \times 5}

Calculating:

T(5)3022e0.443022×0.6443014.1715.83CT(5) \approx 30 - 22 e^{-0.44} \approx 30 - 22 \times 0.644 \approx 30 - 14.17 \approx 15.83^\circ C

Temperature after 10 minutes:

T(10)=3022e0.088×10T(10) = 30 - 22 e^{-0.088 \times 10}

Calculating:

T(10)3022e0.883022×0.415309.1320.87CT(10) \approx 30 - 22 e^{-0.88} \approx 30 - 22 \times 0.415 \approx 30 - 9.13 \approx 20.87^\circ C

Step 4: Graph the relationship between temperature and time

The graph of T(t)=3022e0.088tT(t) = 30 - 22 e^{-0.088t} will show an exponential curve starting at 8C8^\circ C and approaching 30C30^\circ C as tt increases.

To verify these calculations using a graphing tool, plot the function and use TRACE to check the temperatures at t=5t = 5 and t=10t = 10.

Would you like a detailed graph plotted, or any further explanation?


5 Related Questions:

  1. How would the result change if the ambient temperature were different?
  2. What happens to the rate of temperature change as the thermometer's temperature approaches room temperature?
  3. Can you derive the formula for Newton's Law of Cooling from first principles?
  4. How does the constant kk affect the speed of cooling or heating?
  5. 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)