11. The processing of raw sugar has an \"inversion\" step that changes the sugar

Question

11. The processing of raw sugar has an "inversion" step that changes the sugar's molecular structure. Once the process has begun, the rate of change of the amount of raw sugar with respect to time is proportional to the amount of raw sugar remaining. Write a differential equation based on this information by first defining two variables. Then solve the differential equation to find a general solution. Work with this solution to answer the following question: If 1000 kg of raw sugar reduces to 800 kg of raw sugar during the first ten hours, then how much raw sugar will remain after another 14 hours?

Explanation / Answer

Hi,
Let the intial amount of sugat is Y, now given the rate of change is prportional to the amount of remaning
that is
dy/dt y , then we can write,
dy/dt=ky, where k is a constant,
rearranging,
dy/y= k dt
integrating on both sides,
log y=kt+x
i.e y=y0ekt
now, given 1000 reduced to 800 in 10 hours, substituting in above equation,
800=1000 ek*10
i.e ek*10=4/5
k*10= -0.22
k=-0.022
therefore y after 14 more hours is
y=1000 e-0.022*24=589.78 kg
Thumbs up if this was helpful, otherwise let me know in comments

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