Newton's Law of Cooling says that the rate at which a body cools is proportional to the difference in temperature between the body and an environment into which it is introduced. Suppose hot water at a temperature of 175 degrees Fahrenheit is poured into a mug where the room temperature is 70 degrees Fahrenheit. Then, 9 minutes later the temperature of the water is 130 degrees Fahrenheit. Use Newton's law of cooling to find temperature, T, of the water t minutes after it is poured into the mug. Round any constants to four decimal places.
(a) T(t)=
(b) What is the temperature in 35 minutes? Temperature =
degrees Fahrenheit
(c) What is the rate at which the temperature in changing in 35 minutes?
∘F/min
(a) Recall that T(t) = Te + (Te-To)e^(-kt)
T(t) = 70 + (175-70)e^(-kt) = 70 + 105e^(-kt)
since T(9) = 130, find k by solving
70 + 105e^(-9k) = 130
k = 1/9 log(7/4) = 0.062
and so
T(t) = 70 + 105e^(-0.062t)
(b) find T(35)
(c) dT/dt = -6.51 e^(-0.062t)
so find T'(35)
Thanks, 100% correct!
To find the temperature T(t) of the water t minutes after it is poured into the mug, we can use the formula derived from Newton's Law of Cooling:
T(t) = T0 + (Tm - T0) * e^(-kt)
Where:
T(t) is the temperature of the water t minutes after it is poured into the mug
T0 is the initial temperature of the water (175 degrees Fahrenheit)
Tm is the temperature of the surrounding environment (70 degrees Fahrenheit)
k is the cooling constant
(a) Let's find the value of k first using the given information:
We know that after 9 minutes, the temperature of the water has cooled down to 130 degrees Fahrenheit.
So, we can substitute the values into the formula to solve for k:
130 = 175 + (70 - 175) * e^(-9k)
Simplifying the equation:
-45 = -105 * e^(-9k)
Dividing both sides by -105:
e^(-9k) = 9/21
Applying natural logarithm to both sides:
-9k = ln(9/21)
Dividing both sides by -9:
k = ln(9/21) / -9 ≈ -0.0556 (rounded to four decimal places)
Now that we have the value of k, we can find the temperature T(t) at any given time.
(b) To find the temperature in 35 minutes, we substitute the values into the formula:
T(35) = 175 + (70 - 175) * e^(-0.0556 * 35)
Simplifying the formula:
T(35) = 175 + (-105) * e^(-1.944)
Calculating the value using a scientific calculator or software, we find that T(35) ≈ 68.3993 degrees Fahrenheit.
Therefore, the temperature in 35 minutes is approximately 68.3993 degrees Fahrenheit.
(c) To find the rate at which the temperature is changing in 35 minutes, we differentiate the formula with respect to time (t):
dT(t)/dt = k * (Tm - T0) * e^(-kt)
Substituting the values:
dT(t)/dt = -0.0556 * (70 - 175) * e^(-0.0556 * 35)
Simplifying the equation:
dT(t)/dt = -0.0556 * (-105) * e^(-1.944)
Calculating the value using a scientific calculator or software, we find that dT(t)/dt ≈ 3.7168 degrees Fahrenheit per minute.
Therefore, the rate at which the temperature is changing in 35 minutes is approximately 3.7168 degrees Fahrenheit per minute.