Question

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 hours, how much raw sugar will remain after another 14 hours?

Answer 1

Answer:  After 14 hours, 740.74 kg of raw sugar will remain.

Step-by-step explanation:  Given that the rate of change of the amount of raw sugar is proportional to the amount of raw sugar remaining.

Also, 1000 kg of raw sugar reduces to 800 kg of raw sugar during the first 10 hours.

We are to find the amount of raw sugar that will remain after another 14 hours.

Let x and y represents the rate of change of amount of raw sugar and the amount of raw sugar that left respectively.

Then, according to the given information, we have

$x\propto y\\\\\Rightarrow x=ky,~~~[\textup{where k is the constant of proportionality}]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

Now, when y = 800, then x is given by

$x=\dfrac{1000-800}{10}=\dfrac{200}{10}=20.$

So, from equation (i), we have

$20=800k\\\\\Rightarrow k=\dfrac{20}{800}\\\\\Rightarrow k=\dfrac{1}{40}.$

That is,

$x=\dfrac{y}{40}.$

After 14 hours, we have

$\dfrac{1000-y}{14}=\dfrac{y}{40}\\\\\Rightarrow \dfrac{1000-y}{7}=\dfrac{y}{20}\\\\\Rightarrow 20000-20y=7y\\\\\Rightarrow 27y=20000\\\\\Rightarrow y=\dfrac{20000}{27}\\\\\Rightarrow y=740.74.$

Thus, after 14 hours, 740.74 kg of raw sugar will remain.