In this exercise, a 1000-liter tank contains 500 L of water with a salt concentration of 10 g/L. Water with a salt concentration of 50 g/L flows into the tank at a rate of $R_{\text {in }}$=80 L/min. The fluid mixes instantaneously and is pumped out at a specified rate $R_{\text {out. }}$ Let y(t) denote the quantity of salt in the tank at time t.

Assume that $R_{\text {out }}$=40 L/min.

(a) Set up and solve the differential equation for $y(t)$.

(b) What is the salt concentration when the tank overflows?

You have 10 pairs of socks, five black and five blue, but they are not paired up. Instead, they are all mixed up in a drawer. It's early in the morning, and you don't want to turn on the lights in your dark room. How many socks must you pull out to guarantee that you have a pair of one color? How many must you pull out to have two good pairs (each pair is the same color)? How many must you pull out to be certain you have a pair of black socks?

Write the integers from 1 through 100 in 10 lines of 10 numbers each. (a) Cross out the number 1. Cross out all multiples of 2 other than 2 itself. Do the same for 3, 5, and 7. (b) Of what type are the remaining numbers? Explain why this is the only type of number left.

Zoom out sufficiently far so that the graph appears as a line. What equation does this line appear to have? (Note that the points at which the function is not continuous are not readily seen when you zoom out.)

$$f(x)=-\frac{x^{3}-2x^{2}+2}{2x^{2}}$$

$$g(x)= -\frac{1}{2}x+1-\frac{1}{x^{2}}$$

A large tank is filled with water when an outflow valve is opened at $t=0$. Water flows out at a rate, in $\mathrm{gal} / \mathrm{min}$, given by $Q^{\prime}(t)=0.1\left(100-t^2\right)$, for $0 \leq t \leq 10$. a. Find the amount of water $Q(t)$ that has flowed out of the tank after $t$ minutes, given the initial condition $Q(0)=0$. b. Graph the flow function $Q$, for $0 \leq t \leq 10$. c. How much water flows out of the tank in $10 \mathrm{~min}$ ?

In the sentence below, correct any errors in the use of comparisons. Cross out any unnecessary words. Insert any missing word by inserting a caret (^) and writing the word above it. If the sentence is already correct, write C at the end.

Then I carefully cut out the pieces with my dad’s jigsaw, which is the most perfect tool for the job.

Consider a room air conditioner using a Carnot cycle at maximum theoretical efficiency and operating between the temperatures of $18.0^{\circ} \mathrm{C}$ (indoors) and $35.0^{\circ} \mathrm{C}$ (outdoors). For each $1.00 \mathrm{~J}$ of heat flowing out of the room into the air conditioner: a) How much heat flows out of the air conditioner to the outdoors?

A manufacturer of lightbulbs wants to produce bulbs that last about $700$ hours but, of course, some bulbs burn out faster than others. Let $F(t)$ be the fraction of the company's bulbs that burn out before $t$ hours, so $F(t)$ always lies between $0$ and $1$ .

What is the meaning of the derivative $r(t)=F^{\prime}(t)$ ?

A television advertisement shows a lawyer in a bathing suit coming out of a lake. He says, "If you're in over your head because of bad debts, let us bail you out. We're the best firm in the state." Should there be any restrictions on ads like this? If so, what? Should there be other restrictions on ads? If so, what should they be?

Zoom out sufficiently far so that the graph appears as a line. What equation does this line appear to have? (Note that the points at which the function is not continuous are not readily seen when you zoom out.)
$f(x)=-\dfrac{x^3-2 x^2+2}{2 x^2}$

$$g(x)=-\dfrac{1}{2} x+1-\dfrac{1}{x^2}$$

Read each sentence and decide if the underlined number should be spelled out. If it should, write the spelled-out form above it. If the number is already correct, write $C$ above it.

Example 1. $\underline{152}$ (One hundred fifty-two) students auditioned for the marching band.

The temperature should not drop below $\underline{32}$ degrees tonight.

Four blue socks, four white socks, and four gray socks are mixed in a drawer. You pull out two socks, one at a time, without looking.
a. Draw a tree diagram along with the possible outcomes and the probabilities of each outcome.
b. What is the probability of the event of getting a pair of socks of the same color?
c. What is the probability of the event of getting two gray socks?
d. Suppose that, instead of pulling out two socks, you pull out four socks. What is the probability of the event of getting at least two socks of the same color?

Set up the payoff matrix. You must decide whether to attack your opponent by sea or air, and your opponent must simultaneously decide whether to mount an all-out air defense, an all-out coastal defense (against an attack from the sea) or a combined air and coastal defense. If there is no defense for your mode of attack, you win 100 points. If your attack is met by a shared air and coastal defense, you win 50 points. If your attack is met by an all-out defense, you lose 200 points.

A piston $/$ cylinder has $15\ \mathrm{ft}$ of liquid $70 \mathrm{~F}$ water on top of the piston $(m=0)$ with a cross-sectional area of $1\ \mathrm{ft}^2$. Air let in under the piston rises and pushes the water out over the top edge. Determine the work needed to push all the water out and plot the process in a $P-V$ diagram.