Please solve the following question:
The energy of a system as a function of time t...(Please find full question in image.)

Energy of system at time t , E(t) = A² exp( -α t )  .................(1)

 

where α = 0.2 s^-1 is constant 

 

By taking logarithm on both side of eqn.(1) , we get ,  ln [ E(t) ] = 2 ln(A) - α t  ............(2)

 

By differentiating eqn.(2) , we get , ( dE/E ) = 2 (dA/A) - α dt .............(3)

 

we are given that ,  (dA/A ) × 100 = 1.5 %   , i.e., dA/A = 0.015 ................(4)

 

(dt/t) × 100 = 1.25 % , i.e., dt/t = 0.0125   ...................(5)

 

Hence from eqn.(3) , we get at t= 5 s,  ( dE/E )  = 2 × 0.015 + 0.2 × ( .0125 *5 ) =  0.0425

 

( error is always added together , hence - α dt is considered as + α dt  in above calculation.

We can consder error in time ±dt , hence -ve sign is considered  for dt to get maximum error possible)

 

Hence percentage error in E at t= 5 s is 4.25 %