A travel mug of 99 degrees Celsius is left on the roof of a parked car on a cold winter day. The temperature of the coffee after t minutes is given by H=99(2)^(-t/14).

Question

After how many minutes will the coffee be only lukewarm (30 degrees Celsius)?

Answer 1

The 14 in the exponent is the "half-life" of the temperature. When t =14, the mug's temperature will be 99 / 2 = 49.50 degrees, and at t = 28, its temperate will be 49.50 / 2 = 24.75 degrees. So, we know the time to reach 30 degrees is somewhere between 14 and 28 minutes.

To get a more precise measure, we proceed as follows:

30 = 99 * 2^-t/14

30 / 99 = 2^-t/14

Take the logarithm (to any base, but I choose e for this problem) of both sides:

ln ( 30 / 99) =( -t / 14 ) ln(2)

t = -14 * [ ln(1/3.3) / ln(2) ] = 24.1 minutes