Challenge Problem: Newton’s law of cooling says that the temperature, 𝑇(𝑡) at time 𝑡, ofwhich the body is placed in an environment at temperature 𝑇0 is given by𝑇(𝑡) = 𝑇0 + 𝐶𝑒−𝑘𝑡Where 𝐶 and 𝑘 are constants.A piece of metal is heated to 300°C and then placed into a cooling liquid at 50℃. After 4minutes the metal has cooled to 175℃. Find its temperature after 12 minutes.
Question
Challenge Problem: Newton’s law of cooling says that the temperature, 𝑇(𝑡) at time 𝑡, ofwhich the body is placed in an environment at temperature 𝑇0 is given by𝑇(𝑡) = 𝑇0 + 𝐶𝑒−𝑘𝑡Where 𝐶 and 𝑘 are constants.A piece of metal is heated to 300°C and then placed into a cooling liquid at 50℃. After 4minutes the metal has cooled to 175℃. Find its temperature after 12 minutes.
Solution
To solve this problem, we first need to find the values of the constants C and k in the equation T(t) = T0 + Ce^(-kt). We know that the initial temperature of the metal is 300°C (T0), the environment temperature is 50°C, and after 4 minutes (t=4), the temperature of the metal is 175°C.
Step 1: Substitute the known values into the equation:
175 = 50 + C * e^(-4k)
Step 2: Solve for C:
C = (175 - 50) / e^(-4k) = 125 / e^(-4k)
Now we have the value of C in terms of k.
Step 3: To find the value of k, we need more information. Unfortunately, the problem does not provide enough information to find a unique solution for k.
Assuming we had the value of k, we could then find the temperature of the metal after 12 minutes by substituting t=12, T0=50, and the found values of C and k into the equation:
T(12) = 50 + C * e^(-12k)
Without the value of k, it's impossible to find the exact temperature of the metal after 12 minutes.
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