A body of an unknown temperature was placed in a room that was held at 30°F. After 10 minutes, the body's temperature was 0°F, and 20 minutes after the body was placed in the room the body's temperature was 15°F. Estimate the body's initial temperature.
Thank you!!
I am going to assume that the heat flow into the body is proportional to the difference between body temp T and 30 and rate of change of T is proportional to heat flow .
dT/dt = k (30-T)
try T = -ae^-kt + 30
that is 30 at t-->oo and 30-a at t = 0
so
0 = -a e^-10k + 30
15= -a e^-20k + 30
30 = a e^-10k
45 = a e^-20 k
a = 30/e^-10k
45 = 30 (e^-20k/e^-10k)
45/30 = e^-10k
I think you can take it from there, check arithmetic
the 8th line is the simplification of the sixth line,,, im wandering why it is 45? is the algebraic solution wrong?
To estimate the body's initial temperature, we can make use of the concept of thermal equilibrium.
In thermal equilibrium, two objects in contact will eventually reach the same temperature.
Let's consider the body's temperature after 10 minutes. It reached 0°F, which is colder than the room's temperature of 30°F.
Now, let's consider the body's temperature after 20 minutes. It reached 15°F, which is still colder than the room's temperature but now closer to it.
Based on this information, we can infer that the body was cooling down and approaching the room's temperature. We can estimate that the body's initial temperature was likely greater than 15°F, but lower than 30°F.
However, this estimation is based on assumptions and approximations. To get a more accurate estimate, additional data or information would be required.
To estimate the body's initial temperature, we can use the concept of heat transfer. The amount of heat transferred from the body to the surroundings is given by the equation:
Q = m * c * ΔT
Where:
Q is the amount of heat transferred
m is the mass of the body
c is the specific heat capacity of the body's material
ΔT is the change in temperature
In this case, we can assume that the mass of the body remains constant, and the specific heat capacity is also constant. Therefore, we can assume that the amount of heat transferred is directly proportional to the change in temperature:
Q1 / ΔT1 = Q2 / ΔT2
Where:
Q1 is the heat transferred from the body in the first 10 minutes
ΔT1 is the change in temperature of the body in the first 10 minutes
Q2 is the heat transferred from the body in the next 10 minutes
ΔT2 is the change in temperature of the body in the next 10 minutes
In the first 10 minutes, the temperature of the body decreased from an unknown initial temperature to 0°F, so ΔT1 is the initial temperature. In the next 10 minutes, the temperature increased from 0°F to 15°F, so ΔT2 is 15°F.
We can rearrange the equation as follows:
Q1 = (Q2 * ΔT1) / ΔT2
Now we can substitute the available information. We know that Q2 is the amount of heat required to raise the body's temperature from 0°F to 15°F. Therefore, we can write:
Q2 = m * c * ΔT2
Substituting this into the previous equation:
Q1 = (m * c * ΔT2 * ΔT1) / ΔT2
Q2 cancels out in both the numerator and denominator, giving us:
Q1 = m * c * ΔT1
Therefore, the initial temperature can be estimated by finding the ratio of the heat transferred in the first 10 minutes to the change in temperature during that time period.
Given that the initial temperature is T1, the change in temperature during the first 10 minutes is T1 - 0, which simplifies to T1.
Using the information provided, we can now calculate the estimated initial temperature by substituting the values:
Q1 = (m * c * T1) / ΔT2
We know that for the given time intervals, Q1 = Q2. Therefore:
(m * c * T1) / ΔT2 = m * c * ΔT2
The m and c terms cancel out:
T1 / ΔT2 = ΔT2
Cross-multiplying:
T1 = ΔT2^2
Substituting the values:
T1 = (15°F)^2
T1 = 225°F
Therefore, the estimated initial temperature of the body is 225°F.