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  "body": "\n## Example\n\nLet $T(t)$ be the temperature of a body at time $t$. Let $T_a$ be the constant temperature of the surroundings, and let $k \u003e 0$ be a constant.\nNewton's law of cooling is the equation\n$$T'(t) = -k \\left(T(t) - T_a\\right).$$\nWith the initial condition $T(0) = T_0$, its unique solution is\n$$T(t) = T_a + (T_0 - T_a)\\, e^{-k t}.$$\n\n## Remarks\n\nThe difference $D = T - T_a$ satisfies $D' = -k D$. This is [[exponential decay|00SVP6W3CB]], so $D(t) = D(0) e^{-kt}$.\nAs $t \\to \\infty$, $T(t) \\to T_a$. If $T_0 \u003e T_a$ the body cools, and if $T_0 \u003c T_a$ it warms.\nThe law is a good model when the temperature difference is small and the body has a nearly uniform temperature.\n\n## Explanations\n\n### L1 human\n\nA hot cup of coffee cools fast at first and then more slowly.\nThe law says that the speed of cooling is proportional to the difference between the coffee and the room.\nSo the difference halves after each fixed time interval.\n",
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