After heating up in a teapot, a cup of hot water is poured at a temperature of 208, degrees208
∘
F. The cup sits to cool in a room at a temperature of 67, degrees67
∘
F. Newton's Law of Cooling explains that the temperature of the cup of water will decrease proportionally to the difference between the temperature of the water and the temperature of the room, as given by the formula below:
T, equals, T, start subscript, a, end subscript, plus, left bracket, T, start subscript, 0, end subscript, minus, T, start subscript, a, end subscript, right bracket, e, start superscript, minus, k, t, end superscript
T=T
a
+(T
0
−T
a
)e
−kt
T, start subscript, a, end subscript, equalsT
a
= the temperature surrounding the object
T, start subscript, 0, end subscript, equalsT
0
= the initial temperature of the object
t, equalst= the time in minutes
T, equalsT= the temperature of the object after tt minutes
k, equalsk= decay constant
The cup of water reaches the temperature of 191, degrees191
∘
F after 3 minutes. Using this information, find the value of kk, to the nearest thousandth. Use the resulting equation to determine the Fahrenheit temperature of the cup of water, to the nearest degree, after 5.5 minutes.
Enter only the final temperature into the input box.
We are given the initial temperature of the water (T subscript 0) as 208 degrees F, the temperature of the room (T subscript a) as 67 degrees F, and the temperature of the water after 3 minutes (T) as 191 degrees F.
By substituting these values into the formula, we have:
191 = T subscript a + (208 - T subscript a) * e^(-k * 3)
To find the value of k, we can rearrange the equation as follows:
191 - T subscript a = (208 - T subscript a) * e^(-3k)
Dividing both sides by (208 - T subscript a), we get:
(191 - T subscript a) / (208 - T subscript a) = e^(-3k)
Taking the natural logarithm of both sides, we have:
ln((191 - T subscript a) / (208 - T subscript a)) = -3k
Now we can solve for k. Using the given values of T subscript a = 67 and T = 191:
ln((191 - 67) / (208 - 67)) = -3k
ln(124 / 141) = -3k
-0.125 = -3k
k ≈ 0.0417 (rounded to nearest thousandth)
Now we can use this value of k and the formula to find the temperature after 5.5 minutes (t = 5.5):
T = T subscript a + (T subscript 0 - T subscript a) * e^(-kt)
T = 67 + (208 - 67) * e^(-0.0417 * 5.5)
T ≈ 67 + 141 * e^(-0.2294)
T ≈ 67 + 141 * 0.7952
T ≈ 67 + 112.0692
T ≈ 179.0692
Therefore, the temperature of the cup of water after 5.5 minutes is approximately 179 degrees Fahrenheit.