mou2aug
Answered

Suppose you turn off the engine of your car when the temperature of the engine reaches 191 °F. If the outside temperature is a constant 66 °F, then the temperature T of the engine t minutes after you turn off the engine satisfies the equation below.

ln


T − 66

125


= −0.07t

ASolve the equation for T.

T =


BFind the temperature of the engine 30 minutes after you turn it off. Give your answer to the nearest tenth of a degree.

°F

Verified Answer :

antonsandiego
A
Take  Newton’s law of cooling as a basis and make a transformation : 
From : \(ln(T-65/125) = -0.07t\) to :\(ln[ (T - 65 ) / 125 ] = -0.07t\)
Then solve : \(ln[ (T - 65 ) / 125 ] = -0.07t\)
\((T - 65 ) / 125 = e^(-0.07t)\)
And the  result :\(T = 65 + 125e^(-0.07t)\)

b) The easiest way how  you can find the temperature of the engine 30 minutes after you turn it off : \(T = 65 + 125e^(-0.07t)\)
\(T = 65 + 125e^(-0.07*30)\)
Here is the solution in the nearest tenth of a degree  : T = 80.3℉ 

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