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I'm trying to solve #26 from Chapter 7, Transcendental Functions (Thomas' Calculus 12th Edition) and I can't seem to figure out this problem:

The Processing of raw sugar has a step called "inversion" that changes the sugar's molecular structure. Once the process has begun, the rate of change of the amount of raw sugar is proportional to the amount of raw sugar remaining. If 1000 kg of raw sugar reduces to 800 kg of raw sugar during the first 10 hrs, how much sugar will remain after another 1 hours?

I'm guessing this might be just a simple proportion, but I'm not sure. Does it require a differential equation? Perhaps like this:

$$ \frac{dy}{dt}=20t\implies\int{dy}=\int{20t{dt}}\implies{y}=10t^2 $$

$$ \therefore{y}=10\cdot{14}^2=10\cdot{196}=\boxed{1960kg} $$

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2 Answers 2

If the rate is proportional to the amount of sugar remaining, it would be $$\frac{dy}{dt}=ky$$ for some negative constant $k$.

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If, in 10 hours, 80% of the substance is remaining, then, since the rate of inversion is proportional to the amount of sugar left, 1 hour gives you $(0.8)^\frac{1}{10} \approx 0.977933$ or $97.7933 \%$ of the amount from the previous hour.

If you want to find this by solving a differential equation (which you don't need to IMO) then solve the one Michael put up, and then solve for $k$ with the initial condition $y(t = 10 \text{hours}) = 0.8 y(t=0).$

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