At a crime scene, police officers find a murder victim in a tub full of warm water. A thermometer shows that the water temperature is 75 degrees F and the air temperature is 70 degrees F. It is known that most people fill a tub with water at 100 degrees F. Find Newton's Law of cooling and use it to model the temperature of the water in the tub(using a heat transfer coefficient of k=0.018). To the nearest whole minute, how long has the bathtub water been cooling?
I found the equation of T(t)=TA+(TH-TA)e^-kt
75 = 70+(100-70)e^-.018t
t = 99.54
or 100 minutes
To find Newton's Law of Cooling, we first need to understand the equation and the parameters involved. The equation for Newton's Law of Cooling is as follows:
T(t) = TA + (TH - TA) * e^(-kt)
Where:
T(t) is the temperature of the object at time t,
TA is the ambient temperature (air temperature) surrounding the object,
TH is the initial temperature of the object,
k is the heat transfer coefficient, and
e is the base of the natural logarithm (approximately 2.71828).
In this case, we know the following information:
TA (air temperature) = 70 degrees F
TH (initial temperature of the water) = 100 degrees (heat transfer coefficient) = 0.018
We are given that the water temperature at the crime scene is 75 degrees F. We can use this information to find out how long the water has been cooling.
We need to rearrange the equation to solve for time (t):
T(t) = TA + (TH - TA) * e^(-kt)
T(t) - TA = (TH - TA) * e^(-kt)
(T(t) - TA) / (TH - TA) = e^(-kt)
Now, we take the natural logarithm of both sides:
ln[(T(t) - TA) / (TH - TA)] = -kt
Rearranging the equation further, we get:
t = ln[(T(t) - TA) / (TH - TA)] / (-k)
Let's substitute the known values into the equation and solve for t:
t = ln[(75 - 70) / (100 - 70)] / (-0.018)
t = ln[5 / 30] / (-0.018)
t = ln[1/6] / (-0.018)
Using a calculator, we find:
t ≈ 40.9 minutes
Therefore, to the nearest whole minute, the bathtub water has been cooling for approximately 41 minutes.