Question

A metal cube takes 5 minutes to cool from 60 degree celcius to 52 degree celcius . How much time take to cool to 44 degree celcius if the temperature of the surrounding is 32 degree celcius

Answer 1

8/5×8= 12.8
because in 5 minutes the difference in temperature is 8 so pre minute it will be 8/5

Answer 2

Given,

Metal cube cools from 60°C to 52°C in 5 minutes

The temperature of the surroundings, θ$_s$ = 32°C

To find,

Time taken to cool to 44°C

Solution,

We know that

dθ/dt ∝ (θ$_t$ - θ$_s$)

⇒ dθ/dt = k (θ$_t$ - θ$_s$)

when the temperature goes from 60°C to 52°C,

θ$_t$ = (60+52)/2 = 56°C

⇒8/5 = k ( 56 - 32 )

⇒k = 8/5 × 1/24 = 1/15

Now when the temperature goes from 52°C  to 44°C, let the time taken be t.

θ$_t$ = (52+44)/2 = 48°C

dθ/dt = k (θ$_t$ - θ$_s$)

⇒ 8/t = 1/15 (48 - 32)

⇒ t = (8×15)/16

⇒ t = 15/2 = 7.5 minutes

∴ the time taken for the metal cube to cool down to 44°C is 7.5 minutes.