Margo borrows $1400, agreeing to pay it back with 7% annual interest after 13 months. How much interest will she pay? Round your
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a) The initial temperature difference of 75-38 = 37 degrees has been reduced to 60-38 = 22 degrees in 30 minutes. According to the law of cooling, it will be reduced by the same factor of 22/37 in another 30 minutes, so the temperature difference then will be 22(22/37) = 13 degrees.
The temperature of the lemonade after an additional 30 minutes will be 38+13= 51°F.
b) The temperature can be modeled by
T(t) = 38 + 37·(22/37)^(t/30)
Substituting the given information, we have
T(t) = 55 = 38 +37·(22/37)^(t/30)
17/37 = (22/37)^(t/30) . . . . . . subtract 38, divide by 37
log(17/37) = (t/30)·log(22/37) . . . . take the logarithm
t = 30·log(17/37)/log(22/37) . . . . divide by the coefficient of t
t ≈ 44.878
It will take about 45 minutes for the lemonade to cool to 55 °F.
The temperature of the lemonade after an additional 30 minutes will be 38+13= 51°F.
b) The temperature can be modeled by
T(t) = 38 + 37·(22/37)^(t/30)
Substituting the given information, we have
T(t) = 55 = 38 +37·(22/37)^(t/30)
17/37 = (22/37)^(t/30) . . . . . . subtract 38, divide by 37
log(17/37) = (t/30)·log(22/37) . . . . take the logarithm
t = 30·log(17/37)/log(22/37) . . . . divide by the coefficient of t
t ≈ 44.878
It will take about 45 minutes for the lemonade to cool to 55 °F.