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Answer:
A. 118.33 minutes
Step-by-step explanation:
You want to know the time it takes to cool to 50°F in an environment of 30°F, starting from an initial temperature of 180°F, if cooling to 120°F occurs in 30 minutes.
Newton's law of cooling tells you the rate of change of temperature is proportional to the difference from the final temperature. That means the temperature curve is exponential. We can find its parameters based on the given information.
Let's define the parameters as follows:
[tex]T_0=\text{initial temperature}\\\\T_f=\text{final temperature}\\\\T_1=\text{waypoint temperature}\\\\t_1=\text{waypoint time}[/tex]
Using these parameters, we can write the equation for the temperature as a function of time: T(t).
[tex]T(t)=T_f+(T_0-T_f)\left(\dfrac{T_1-T_f}{T_0-T_f}\right)^{\dfrac{t}{t_1}}\\\\\\T(t)=30+(180-30)\left(\dfrac{120-30}{180-30}\right)^{\dfrac{t}{30}}=30+150(0.6)^{t/30}[/tex]
We want to find the value of t for which T(t) is 50.
[tex]50 = 30 +150(0.6)^{t/30}\\\\\dfrac{20}{150}=0.6^{t/30}\\\\\\\log\left(\dfrac{2}{15}\right)=\dfrac{t}{30}\log(0.6)\\\\\\t=\dfrac{30\log\left(\dfrac{2}{15}\right)}{\log(0.6)}\approx 118.33[/tex]
It will take about 118.33 minutes for the custard to cool to 50°F, choice A.