Question 1189396
the engine is 87 degrees centigrade when it is shut off.


newton's law of cooling is:


T(t) = Ts + (T0 - Ts) * e ^ -kt.


T(t) is equal to 33 degrees.
T0 is equal to 87 degrees.
Ts is equal to 15 degrees.


the formula becomes:


33 = 15 + (87 - 15) * e ^ (-.0347 * t)


simplify to get:


33 = 15 + 72 * e ^ (-.0347 * t)


subtract 15 from both sides of the equation to get:


18 = 72 * e ^ (-.0347 * t)


divide both sides of the equation by 72 to get:


18/72 = e ^ (-.0347 * t)


take the natural log of both sides of the equation to get:


ln(18/72) = ln(e ^ (-.0347 * t)


by laws of logs, this becomes:


ln(18/72) = -.0347 * t * ln(e).


since ln(e) = 1, this becomes:


ln(18/72) = = -.0347 * t


divide both sides of the equation by -.0347 to get:


ln(18/72) / -.0347 = t


solve for t to get:


t = 39.95084614.


since since 39.95... rounded to 40, your solution is that t = 40 minutes.


the online calculator at <a href = "https://www.calctown.com/calculators/newton-law-of-cooling" target = "_blank">https://www.calctown.com/calculators/newton-law-of-cooling</a> give the same result.


here's a display of the results from using that calculator.


<img src = "http://theo.x10hosting.com/2022/010801.jpg" >