Mike cooked some steak, and when he took it off the stove, the internal temperature of the steak was \(135 \degree \text{F}\). The temperature of the room in which Mike put the steak to cool down was \(70\degree \text{F}\). He measured the internal temperature of the steak every 10 minutes and recorded his observations in the table below.

$$ \begin{array} {|c|c|} \hline \text{Time (minutes)} & \text{Temperature} \\ \hline 0 & 135\degree \text{F} \\ \hline 10 & 94\degree \text{F} \\ \hline 20 & 82\degree \text{F} \\ \hline 30 & 75\degree \text{F} \\ \hline 40 & 73\degree \text{F} \\ \hline 50 & 72\degree \text{F} \\ \hline \end{array} $$

Which of the following best models the relationship between the time \(t\), in minutes, since the food was removed from the stove and the temperature \(T\), in degrees Fahrenheit, of the steak?

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Since the rate of decrease of the temperature is not constant per 10 minutes, our model cannot be linear. It appears that the temperature decreases sharply first, but slows down as the temperature approaches room temperature.

The only difference between the exponential models is the coefficient of the exponential portion. We may be tempted to pick \(135\), but be careful. We want to find a model that gets us \(T=135\) when \(t=0\).

Realistically, the internal temperature of steak usually rises a bit after it is taken off the stove. The heat from the outer layers transfer to the inner layers of the steak, raising the doneness of a steak. The steak in our question would probably be medium \((140\degree \text{F} )\) rather than medium rare \((135\degree \text{F} )\) after resting in a covered container for approximately 10 minutes.

The external temperature would likely follow a similar exponential model.