A thermometer is removed from a room where the temperature i

A thermometer is removed from a room where the temperature is 70

Solution

T(t) = Ta + (Th - Ta)*e^(-kt)

where,

T(t) = Temperature at time t,
TA = Ambient temperature (temp of surroundings),
TH = Temperature of hot object at time 0,
k = positive constant and
t = time.

T(t) = 10 + (70 - 10)e^(-kt)

T(t) = 10 + 60e^(-kt)

When t = 0.5 , T(t) = 40 :

40 = 10 + 60e^(-k*0.5)

30 = 60e^(-k*0.5)

0.5 = e^(-k*0.5)

ln(0.5) = -0.5k

k = -2ln(0.5) = 1.38629436

So, T(t) = 10 +60e^(-kt)
now becomes
T(t) = 10 + 60e^(- 1.38629436t)

At t = 1 :

T(1) = 10 + 60e^(- 1.38629436 * 1) = 25 ---> ANSWER

15 :

15 = 10 + 60e^(- 1.38629436t)

5 = 60e^(- 1.38629436t)

1/12 = e^(- 1.38629436t)

ln(1/12) = - 1.38629436t

t = 1.79 minutes ---> ANSWER