Question

A car engine runs at a temperature of 190 °F. When it is turned off it cools according to Newton's Law of Cooling with constant k = 0.0341, with the time measured in minutes. Find the time needed for the engine to cool to 90 °F if the surrounding temperature is 60 °F.

Answer 1

To find the time needed for the car engine to cool to 90 °F, we can use Newton's Law of Cooling with the given temperatures and cooling constant. By rearranging the equation and solving for time, we find that it will take approximately 109.73 minutes for the engine to cool to 90 °F.

Step-by-step explanation:

To find the time needed for the car engine to cool to 90 °F, we can use Newton's Law of Cooling. The formula for Newton's Law of Cooling is: T(t) = Ts + (T0 - Ts)e-kt, where T(t) is the temperature at time t, Ts is the surrounding temperature, T0 is the initial temperature, k is the cooling constant, and t is the time in minutes.

Given that the initial temperature (T0) is 190 °F, the surrounding temperature (Ts) is 60 °F, and k is 0.0341, we can plug these values into the formula. Letting T(t) be 90 °F, we can solve for t by rearranging the equation and taking the natural logarithm of both sides:

ln((90 - 60) / (190 - 60)) = -0.0341t

Solving for t using a calculator or software, we find that t ≈ 109.73 minutes. Therefore, it will take approximately 109.73 minutes for the engine to cool to 90 °F.