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?
To find Newton's Law of cooling and model the temperature of the water in the tub, we can use the following formula:
dT/dt = -k * (T - Ta),
where dT/dt is the rate of change of temperature with respect to time, T is the temperature of the water, Ta is the temperature of the surrounding air, and k is the heat transfer coefficient.
In this case, we have:
Initial temperature of the water (T0): 100 degrees F
Temperature of the surrounding air (Ta): 70 degrees F
Temperature of the water at the crime scene (T): 75 degrees F
Heat transfer coefficient (k): 0.018
Substituting the values into the formula, we can solve for the time (t) it took for the water to cool from 100 degrees F to 75 degrees F.
dT/dt = -k * (T - Ta)
dT/dt = -0.018 * (75 - 70)
dT/dt = -0.018 * 5
dT/dt = -0.09
To find the time t, we need to integrate the above equation:
∫ dT / (T - Ta) = ∫ -0.09 dt
Integrating both sides, we get:
ln |T - Ta| = -0.09t + C,
where C is the constant of integration.
To find the value of C, we can use the initial condition T = T0 at t = 0:
ln |T0 - Ta| = C.
Therefore, the equation becomes:
ln |T - Ta| = -0.09t + ln |T0 - Ta|.
Now, we can substitute the given values and solve for t:
ln |75 - 70| = -0.09t + ln |100 - 70|
ln |5| = -0.09t + ln |30|
ln 5 = -0.09t + ln 30.
By solving this equation for t, we get:
t ≈ -23.04 minutes.
However, time cannot be negative in this context, so we ignore the negative value. Therefore, to the nearest whole minute, the bathtub water has been cooling for approximately 23 minutes.
To find Newton's Law of cooling and use it to model the temperature of the water in the tub, we need to understand the law and its formula. Newton's Law of cooling states that the rate of change of temperature of an object is proportional to the difference between the object's temperature and the ambient temperature.
The general form of Newton's Law of cooling is as follows:
dT/dt = -k (T - Ta)
Where:
dT/dt is the rate of change of temperature with respect to time,
T is the temperature of the object,
Ta is the ambient temperature, and
k is the heat transfer coefficient.
In our case, the temperature of the water in the tub is initially 100 degrees F (T0 = 100), the ambient temperature is 70 degrees F (Ta = 70), and the heat transfer coefficient is given as k = 0.018. We need to find the time it takes for the temperature of the water to decrease from 100 degrees F to 75 degrees F.
Let's substitute the given values into the formula:
(1) dT/dt = -0.018 (T - 70)
To solve this first-order linear differential equation, we can rearrange it and solve by separation of variables.
Divide both sides by (T - 70):
(2) dT / (T - 70) = -0.018 dt
Now integrate both sides of the equation:
∫ dT / (T - 70) = ∫ -0.018 dt
Applying the integral:
ln|T - 70| = -0.018t + C
Where C is the integration constant.
To find C, substitute the initial condition T(0) = 100 and solve for C:
ln|100 - 70| = -0.018(0) + C
ln(30) = C
The equation becomes:
ln|T - 70| = -0.018t + ln(30)
Now we can solve for t when T = 75:
ln|75 - 70| = -0.018t + ln(30)
ln(5) = -0.018t + ln(30)
Rearranging the equation:
0.018t = ln(30) - ln(5)
0.018t = ln(30/5)
0.018t = ln(6)
Divide both sides by 0.018:
t = ln(6) / 0.018 ≈ 9.501
Since we are asked to give the answer to the nearest whole minute, the approximate time it has been cooling is 10 minutes.