document.write( "Question 1032283: Newton’s law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. So the rate of cooling for a bottle of lemonade at a room temperature of 75°F which is placed into a refrigerator with temperature of 38°F can be modeled by dT/dt=k(T-38) where T(t) is the temperature of the lemonade after t minutes and T(0) = 75. After 30 minutes the lemonade has cooled to 60°F, so T(30) = 60.
\n" ); document.write( "To the nearest minute, how long does it take for the lemonade to cool to 55°F? \n" ); document.write( "

Algebra.Com's Answer #646937 by Fombitz(32388) $\"\"$ $\"About$

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Substitute,
\n" ); document.write( " $\"u=T-38\"$
\n" ); document.write( " $\"du=dT\"$
\n" ); document.write( "So,
\n" ); document.write( " $\"du%2Fdt=ku\"$
\n" ); document.write( " $\"du%2Fu=kdt\"$
\n" ); document.write( " $\"ln%28u%29=kt%2BC\"$
\n" ); document.write( " $\"u=e%5E%28kt%2BC%29\"$
\n" ); document.write( " $\"u=Ce%5E%28kt%29\"$
\n" ); document.write( " $\"T-38=Ce%5E%28kt%29\"$
\n" ); document.write( "So when $\"t=0\"$ ,
\n" ); document.write( " $\"T=75\"$
\n" ); document.write( " $\"T-38=C%281%29\"$
\n" ); document.write( " $\"C=75-38\"$
\n" ); document.write( " $\"C=75\"$
\n" ); document.write( "and
\n" ); document.write( "when $\"t=30\"$ ,
\n" ); document.write( " $\"T=60\"$
\n" ); document.write( " $\"60-38=75e%5E%2830k%29\"$
\n" ); document.write( " $\"e%5E%2830k%29=%2860-38%29%2F75\"$
\n" ); document.write( " $\"e%5E%2830k%29=22%2F75\"$
\n" ); document.write( " $\"30k=ln%2822%2F75%29\"$
\n" ); document.write( " $\"k=%281%2F30%29ln%2822%2F75%29\"$
\n" ); document.write( "So now the equation is complete.
\n" ); document.write( ".
\n" ); document.write( ".
\n" ); document.write( ".
\n" ); document.write( "Find $\"T\"$ when $\"t=60\"$
\n" ); document.write( " $\"T-38=75e%5E%2860k%29\"$
\n" ); document.write( " $\"T-38=75e%5E%282%2Aln%2822%2F75%29%29%7D%7D%0D%0A%7B%7B%7BT-38=6.45%29\"$
\n" ); document.write( " $\"highlight%28T=44.5%29\"$ $\"F\"$
\n" ); document.write( ".
\n" ); document.write( ".
\n" ); document.write( ".
\n" ); document.write( "Find $\"t\"$ when $\"T=55\"$
\n" ); document.write( " $\"55-38=75e%5E%28kt%29\"$
\n" ); document.write( " $\"e%5E%28kt%29=17%2F75\"$
\n" ); document.write( " $\"kt=ln%2817%2F75%29\"$
\n" ); document.write( " $\"%281%2F30%29ln%2822%2F75%29t=ln%2817%2F75%29\"$
\n" ); document.write( " $\"t=%2830ln%2817%2F75%29%29%2Fln%2822%2F75%29\"$ $\"min\"$
\n" ); document.write( " $\"highlight%28t=36.3%29\"$ $\"min\"$
\n" ); document.write( " \n" ); document.write( "

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