You place a cup of 205°F coffee on a table in a room that is 72°F, and 10 minutes later, it is 195°F. Approximately how long will it be before the coffee is 180°F?

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You place a cup of 205°F coffee on a table in a room that is 72°F, and 10 minutes later, it is 195°F. Approximately how long will it be before the coffee is 180°F?

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Answer 1 · 2017-05-20T21:34:41

Approximately $27$ $27$ minutes.

Explanation:

This is a Newton's Law of Cooling Problem.

Newton's Law of Cooling states that an object cools down by the formula $T (t) = T_{s} + (T_{0} - T_{s}) e^{- k t}$ $T(t) = T_s + (T_0 - T_s)e^(-kt)$ , where $t_{0}$ $t_0$ is the initial temperature of the liquid, $t_{s}$ $t_s$ is the temperature of the surrounding environment, $t$ $t$ is the number of minutes elapsed and $T (t)$ $T(t)$ is the temperature after $t$ $t$ minutes elapsed. $k$ $k$ is the constant, which will differ depending on the object.

We have to start by finding $k$ $k$ .

$195 = 72 + (205 - 72) e^{- k (10)}$ $195 = 72 + (205 - 72)e^(-k(10))$

$\frac{123}{133} = e^{- 10 k}$ $123/133 = e^(-10k)$

$ln (\frac{123}{133}) = - 10 k$ $ln(123/133) = -10k$

$k = - \frac{1}{10} ln (\frac{123}{133})$ $k = -1/10ln(123/133)$

So our formula is

$T (t) = 72 + 133 e^{\frac{1}{10} ln (\frac{123}{133}) t}$ $T(t) = 72 + 133e^(1/10ln(123/133)t)$

We're looking for the amount of time it takes for the coffee to cool to $180$ $180$ degrees fahrenheit, so we write the equation:

$180 = 72 + 133 e^{\frac{1}{10} ln (\frac{123}{133}) t}$ $180 = 72 + 133e^(1/10ln(123/133)t)$

$\frac{108}{133} = e^{\frac{1}{10} ln (\frac{123}{133}) t}$ $108/133 = e^(1/10ln(123/133)t)$

$ln (\frac{108}{133}) = \frac{1}{10} ln (\frac{123}{133}) t$ $ln(108/133) = 1/10ln(123/133)t$

A good approximation gives

$t \approx 26.64$ $t ~~ 26.64$

Therefore, it will take approximately $27$ $27$ minutes for the coffee to cool to $180˚$ F.

Hopefully this helps!

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You place a cup of 205°F coffee on a table in a room that is 72°F, and 10 minutes later, it is 195°F. Approximately how long will it be before the coffee is 180°F?

1 Answer

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