A hot metal bar is submerged in a large reservoir of water whose temperature is

Question

A hot metal bar is submerged in a large reservoir of water whose temperature is 70F. The temperature of the bar 20s after submersion is 105F. After 1min submerged, the temperature has cooled to 85F.

(a) Determine the cooling constant k . k=_____ s^1

(b) What is the differential equation satisfied by the temperature F(t) of the bar? F(t)= _______ (Use F for F(t))

(c) What is the formula for F(t)? F(t)= ______

(d) Determine the temperature of the bar at the moment it is submerged. Initial temperature = ______F

Explanation / Answer

  (a)From the question we can understand that the temperature here is a decaying exponential, so

T(t) = 70 + ( T1 - 70) exp(-k t) where t is the time measured in seconds, and T1 is the initial temperature of the bar.

so we have T(20) = 70 + (T1 - 70) exp(-20k) = 105, and

T(60) = 70 + (T1 - 70) exp(-60k) = 85,

60 comes because it is mentioned that after 1 min i.e 60 s

Therefore, with some simplification we have

(T1 - 70) exp(- 20 k) = 35, and

(T1 - 70) exp(- 60 k) =15

Dividing the two equations, we get exp(-20k) / exp(-60 k) = 35/15 = 2.33

but exp(-20 k) / exp(-60 k) = exp(-20k + 60 k) = exp( 40 k), therefore,

exp(40 k) =2.33, by taking the ln( ) of both sides, we get 40 k = ln(2.33),

so k = ln (1.8) / 40 = 0.0211 s-1

(b) The differential equation is d F / dt = k ( F0 - F ),

where F is the temperature (in Fahrenheit) of the bar and

F0 is the temperature (in Fahrenheit) of the reservoir water.

This is Newton's law of cooling.

(c) The formula for F(t) is as mentioned in (b), or we can solve it using standard techniques of differential equations to obtain (by undetermined coefficient method)

F (t) = A exp(-k t) + B, the initial condition is F = F1 the initial temperature of the bar.

Note that d F / d t = - k (A exp(kt) ) = k ( - F + B ), so B = F0 the temperature of the reservoir.

i.e. F(t) = A exp(-kt) + F0, and since the initial temperatue is F(0) = F1 , then

F1 = A exp(0) + F0 , that is, A = F1 - F0, so

F(t) = F0 + (F1 - F0) exp(- k t)

(d) We have F(t) = 70 + (F1 - 70) exp( - k t)

and we know that F(20) = 105 = 70 + (F1 - 70) exp( - 20 k) = 70 + (F1 - 70) exp(-20(0.0211))

so 105 = 70 + (F1 - 70)*0.3678 ,

so (F1 - 70) = 35 / 0.3678 =95.139

So F1 = 70 + 95.139 =165.1390F

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