Answer :

**Given:**Temperature of the water : T

_{1}= 50 °C

Height of the vessel : h = 10 cm = 0.1 m

Cross section area of the vessel: A = 10 cm

^{2}= 0.001 m

^{2}.

Thickness of the flat parts: x = 1mm = 0.001 m

Thermal conductivity of the aluminium: K =200 J s

^{–1}m

^{–1}°C

^{–1}

Temperature outside: T

_{2}= 20 °C

Density of water : ρ = 1000 kg m

^{-3}

The specific heat capacity of water = 4200 J k

^{–1}m

^{–1}°C

^{–1}.

Since the walls are adiabatic, no heat transfer would take place. Hence the heat would be transferred only via flat surfaces which are Up and Bottom surfaces.

**Formula used:**

Rate of amount of heat flowing is given as:

Here, Δθ is the amount of heat transferred, ΔT is the temperature difference, K is the thermal conductivity of the material, A is the area of cross section of the material and x is the thickness of the material.

Rate of flow of heat from both the flat surfaces would be:

Hence, rate of heat flow from both the flat surfaces is 12000 J/s.

We know that,

ΔQ=Δθ = msΔT

Where, ΔQ is the change in heat energy, m is the mass , s is the specific heat of the substance and ΔT is the change in temperature for the substance.

Also,

Mass = Density × Volume

∴ m = 1000 × 0.001× 0.1

∴ m = 0.1 kg

We need to find time taken for the temperature to fall by 1°C

Thus, ΔT = 1°C

Substituting in equation (2) and using (1) we get,

Hence, it took around 0.035 seconds for the temperature to drop by 1°C.

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