Problem $1$: $A270$ gallon tank initially contains $200$ gallons $of$ pure water. Water with a concentration $of\frac{?}{(c\frac{?}{?})}$ pounds $of$ salt per gallon $is$ added $to$ the tank $at\frac{?}{(2\frac{?}{\text{m}\text{a}\text{t}\text{h}\text{r}\text{m}\{gal\}}\frac{?}{?}\frac{?}{\text{m}\text{a}\text{t}\text{h}\text{r}\text{m}\{min\}}\frac{?}{?})},$ and the resulting solution leaves $at$ a rate $of\frac{?}{(1\frac{?}{\text{m}\text{a}\text{t}\text{h}\text{r}\text{m}\{gal\}}\frac{?}{?}\frac{?}{\text{m}\text{a}\text{t}\text{h}\text{r}\text{m}\{min\}}\frac{?}{?}).}$ What concentration $\frac{?}{(c\frac{?}{?})}$ should $be$ used $so$ that the tank will have a concentration $of\frac{?}{(\frac{?}{\text{f}\text{r}\text{a}\text{c}\{1\}\{2\}}\frac{?}{?})}$ pounds per gallon once $itis$ full with $270$ gallons $of$ water $in$ the tank?

Problem $2$: Consider the following equation for a certain population $of$ invasive fish $ina$

lake given $by\text{'}P(t)\text{'}(\text{'}t\text{'}is$ measured $in$ years$)$: $\frac{dP}{dt}=2\mathrm{.}4*P(1-\frac{P}{8})(\frac{P}{4}-1)$

$a.To$ account for this fishing, the population model can $be$ updated $to$:

$\frac{dP}{dt}=2\mathrm{.}4*P(1-\frac{P}{8})(\frac{P}{4}-1)$

What values $of$ that can $be$ chosen $to$ ensure that the invasive fish species will $be$ com$-$

pletely removed from the lake $(regardlessof$ the starting population$).$

Hint: Consider the equilibrium points, phase line, and long term behavior $in$ relation $to$

$$ Problem $3$: The amount $of$ money $inan$ account, $S(t),$ which starts with $$S_o$ and grows

with a continuous annual interest rate $ofr$ and continuous deposits $of$ $$D$ per year can $be$

modelled $by$ the initial value problem:

$\frac{dS}{dt}=r*S+D,S(0)=S_o$

Part $a$$$2500000,$ which they put into

$an$ investment account $at$ age $22$ and earns a continuous annual interest rate $of8\%on$ their

investments.

The student wishes $to$ have have $$20000000at$ the time that they retire. How much money

can they continuously withdraw each year and still have $$20000000at$ the time that they

retire $at$ age $60?$

Note: $In$ the case $of$ withdrawls,

Part $b{D}_2$ that someone needs $to$ make $in$ yearly contributions starting $at$

age $22to$ retire with the same amount $of$ money age $60,$ assuming the same $8\%$ continuous

annual interest rate $on$ their investment but with $an$ initial deposit $of$ $$0.$ Problem $4$: Find all functions $M(x,y)$ such that the differential equation:

$M(x,y)+(4*x^4*y^3+x*cos(y))(\frac{dy}{dx})=0$

$is$ exact.