A cool object is to be heated to a maximum temperature M = M°C. At any time t, the rate at which the temperature rises is proportional to the difference between the actual temperature and the maximal temperature. If the object is originally 0°C, find and solve a differential equation describing this situation. Is this the solution: $y ( t ) = M - M e ^ { - k t }$

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Quizplus · Accepted Answer

The given situation describes a process of heating where the rate of temperature increase is proportional to the difference between the current temperature and the maximum temperature, which can be modeled by the differential equation $\frac{dy}{dt} = k(M - y)$. Solving this differential equation indeed yields the solution $y(t) = M - Me^{-kt}$, where $y(t)$ represents the temperature at time $t$, $M$ is the maximum temperature, and $k$ is a positive constant representing the rate of heating.

A Cool Object Is to Be Heated to a Maximum y(t)=M−Me−kty ( t ) = M - M e ^ { - k t }y(t)=M−Me−kt

A Cool Object Is to Be Heated to a Maximum $y ( t ) = M - M e ^ { - k t }$