Question $1$

$1$ pts

A $200-$gallon tank initially contains $200$ gallons of pure water. Suppose that sugar is added to the tank at a rate of $2$ pounds per minute, that sugar water is removed at a rate of $5$ gallons per minute, and we constantly add pure water to keep the tank full. The water in the tank is kept well$-$mixed. Set up a differential equation to study the amount of the sugar in pounds in the tank as a function of $t$ in minutes.

Let $y$ represent the amount of sugar $($in pounds$)$ in the water $($at any given time$).$

Then the rate of change of $y($in pounds per minute$),$ which is $\frac{dy}{dt},$

is the difference between the rate of adding sugar and the rate of removing sugar.

The rate of adding sugar is $2$ pounds per minute.

The rate of removing sugar is the rate of removing solution $($in pounds per minute$)$ multiplied by the density $($in pounds per gallon$)$ of the sugar in the solution.

The density of sugar in the tank $($at any given time$)$ is $\frac{y}{200}($in pounds per gallon$).$

We therefore have the differential equation that relates $\frac{dy}{dt}$ and $y.\frac{dy}{dt}=2-5(\frac{y}{200}).$

True

False