Question

You taken ice-cream out of the freezer,kept at - 18°C. Outside it is 32°C After one minute, the ice cream has warmed to - 8°C what is the temperature of ice cream ofter five minutes​

Answer 1

Explanation:

Let T be the temperature of the ice-cream (in ∘C) after t minutes out of the freezer. Then Newton's law gives

dTdt=−k(T−32)=−kT+32k.

Remembering that k is a constant, solving this differential equation gives a general solution of

T=Ce−kt+32.

Since T=−18 when t=0, we obtain −18=C+32, so C=−50. Since T=−8 when t=1, we have −8=−50e−k+32, which gives k=loge54. We obtain

T=−50e−(loge54)t+32=−50(45)t+32.

(Here we used some index laws.) Hence, when t=5, we have

T=−50(45)5+32=1952125≈15.6 ∘C.

Your ice-cream has well and truly melted! Note that we have effectively assumed that the ice-cream is a block of a single temperature — not very realistic.

Answer 2

Answer:

Let T be the temperature of the ice cream (in °C) after t minutes.

Then, according to the Newton's Law of Cooling, we have

dT/dt = -k (T - T₀)             ---------------------   (i)

where dT/dt = Rate of cooling

k = Constant

T = Temperature of the ice cream

T₀ = Temperature of the surroundings

Taking (i), we have

Her T₀ = 32°C

dT/dt = -k (T - 32)  

dT/(T-32) = -k dt

Integrating both sides, we get,

∫[dT/(T-32)] = ∫-k dt

log (T - 32) = -kt + c

T - 32 = $e^{-kt + c}$

T - 32 = $e^{-kt} . e^c$

Let $e^c$ = C.

T - 32 = $e^{-kt}$C

T = C$e^{-kt}$ + 32             ---------------------- (ii)

Applying initial conditions, we have

T = -18°C at t = 0 min

-18 = Ce⁰ + 32

-18 = C(1) + 32

-18 = C + 32

C = -18 - 32

C = -50

T = -8°C at t = 1 min

-8 = C$e^{-k(1)}$ + 32

But C = -50

Now, we have,

-8 = -50$e^{-k}$ + 32

Solving this, we get

k = log (5/4)

Putting the values of C and k in (ii), we have,

T = -50$e^{-log(5/4)t$ + 32

T = -50$e^{log(4/5)t}$ + 32

T = -50$e^{log(4/5)^t}$ + 32

T = -50 $(\frac{4}{5} )^t$ + 32

Given that time is five minutes, i.e., t = 5 min

T = -50 $(\frac{4}{5} )^5$ + 32

T = 1952/125

T = 15.616°C

Hence, the temperature of ice cream after five minutes​ is T = 15.616°C.

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