Ordinary Differential Equations

The rate at which the ice melts is proportional to the amount of ice at the instant. Find the amount of ice left after 2 hours if half the quantity melts in 30 minutes. Solution. Let m be the amount of ice at any time t.

Solutions

Expert Solution

∴dmdt=km⇒dmm=kdt∫dmm=k∫dt+C⇒log⁡m=kt+C At t=0,m=M:log⁡M=0+C⇒C=log⁡M On putting the value of C, (1) becomes, log⁡m=kt+log⁡Mm=M2 when t=12 hour log⁡M2=k2+log⁡M⟹log⁡M2M=k2⟹log⁡12=k2 or k=2log⁡12 On putting the value of k in (2), we have log⁡m=(2log⁡12)t+log⁡M On putting t=2 hours in (3), we have log⁡m=4log⁡12+log⁡M⇒log⁡mM=log⁡(12)4 or mM=116 or m=M16 After 2 hours, amount of ice left \( \frac{m}{M}=\frac{1}{16} \)of the amount of ice at the beginning.

Therefore. \( \frac{m}{M}=16 \)

SUN BUNRA answered 1 year ago

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