A body at temperature 40°C is kept in a surrounding of constant temperature 20°C. It is observed that its temperature falls to 35°C in 10 minutes. Find how much more time will it take for the body to attain a temperature of 30°C.

Explanation

`Delta theta_f=Delta theta_i e^(-kt)`
for the interval in which temperature falls from 40 to `35^@ C`
`(35-20)=(40-20) e^(-k.10)`
`rArr e^(-10 k) =(3)/(4)`
`rArr K=(ln (4)/(3))/(10)`
For the next interval
`(30-20)=(35-20) e^(-kt)`
`rArr e^(-10) k = (2)/(3)`
`rArr kt= ln (3)/(2)`
`rArr ((ln (4)/(3))t)/(10)=ln (3)/(2)`
`rArr t=10 ((ln(3)/(2)))/((ln (4)/(3)))` minute = 14.096 min
Aliter : (by approximate method)
for the interval in which temperature falls from 40 to `35^@ C`
` lt theta gt = (40+35)/(2) = 37.5^@ C`
from equation (14.4) ` langle (d theta)/(dt) rangle=-k( lt theta gt- theta_0)`
`rArr ((35^@ C - 40^@ C))/(10 (min))=-K (37.5 ^@ C - 20^@ C)`
`rArr K=(1)/(35)(min^(-1))`
for the interval in which temperature falls from `35^@ C to 30^@ C`
`lt theta gt = (35+ 30)/(2) = 32.5^@ C`
from equation (14.4)
`((30^@ C- 35^@ C))/(t)=-(32.5^@ C-20^@ C)`
`rArr` required time,
`t=(5)/(12.5)xx35 "min" = 14 "min"`