document.write( "Question 1053355: Logarithm Function:
\n" ); document.write( "A thermometer marks a reading of 79 degrees fahrenheit, located inside a storage room that has a constant cold temperature reading of 37 degrees fahrenheit.If the thermometer reads 74 degrees fahrenheit in 13 mins, how much time would you have to wait until it reaches a temperature of 54 degrees fahrenheit? (Assuming that the cold follows the laws of newton. Circle the answer to the nearest whole minute). \n" ); document.write( "

Algebra.Com's Answer #668625 by htmentor(1343) $\"\"$ $\"About$

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Newton's law of cooling can be written as T(t) = Ta + (T0-Ta)*exp(-kt),
\n" ); document.write( "where Ta = ambient temperature, T0 = initial temperature and k is a constant
\n" ); document.write( "In this problem, T0 = 79 degrees (the initial temperature) Ta = 37 degrees (constant temperature)
\n" ); document.write( "So the general formula for temperature vs. time is: T(t) = 37 + (79-37)exp(-kt) = 37 + 42*exp(-kt)
\n" ); document.write( "The temperature after 13 mins is 74 deg:
\n" ); document.write( "T(13) = 37 + 42*exp(-13k)
\n" ); document.write( "Solve for k: k = -ln(37/42)/13 = 0.00975
\n" ); document.write( "So the temperature equation is T = 37 + 42*exp(-0.00975t)
\n" ); document.write( "54 = 37 + 42*exp(-0.00975t)
\n" ); document.write( "Solve for t:
\n" ); document.write( "ln(17/42)/-0.00975 = t = 93 mins.
\n" ); document.write( "The temperature graph looks like:
\n" ); document.write( " $\"+graph%28+300%2C+200%2C+-10%2C+100%2C+-10%2C+90%2C+37%2B42%2Aexp%28-0.00975%2Ax%29%29+\"$ \n" ); document.write( "

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