Frank’s automobile engine runs at 100∘c. one day, when the outside temperature is 15∘c, he turns off the ignition and notes that 5 minutes later, the engine has cooled to 70∘c. when will the engine cool to 40∘c? (use decimal notation. give your answer to the nearest whole number.)

the engine cool to 40 at 14.07 minutes

Explanation:

Given information

T(5) = 70

= 100

C = 15

Newton's law of cooling :

T(t) = C + ( - C)

where

T(t) = temperature at any given time

C = surrounding temperature

= initial temperature of heated object

k = cooling constant

to find the the time when the engine will be cooled down to 40, we first need to find the cooling constant, k

when t = 5, T(5) = 70

so,

T(t) = C + ( - C)

T(5) = 15 + (100 - 15)

70 = 15 + (85)

= (70 - 15) / 85

-5k = ln (55/85)

k = - ln (55/85) / 5

k = 0.087

thus, we have the eqaution

T(t) = 15 + (85)

now we can determine the time when T(t) = 40

40 = 15 + (85)

= (40-15)/85

-0.0087t = ln (25/85)

t = - ln (25/85)/0.087

t = 14.07 minutes

see earlier answer

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