An aeration basin is used to oxygenate wastewater at 283 K. The basin is filled to a depth of 4.55 m with $283 \mathrm{m}^{3}$ wastewater of 0.050 mmole/L initial dissolved oxygen $\left(\mathrm{O}_{2}\right)$ cocentration. The filled basin is then aerated with six six spargers, with each sparger delivering compressed air to bottom of the basin at a volumetric flow rate of $7.08 \times 10^{-3} \mathrm{m}^{3} / \mathrm{s}$. The dissolved solids content will be low enough that Henry's law will be obeyed, with a Henry's law constant of $H=3.27 \times 10^{4} \mathrm{atm}$ $\left(p_{A}=H \cdot x_{A}^{\circ}\right)$ at 283 K. For purposes of this calculation, it may be assumed that the rate of $\mathrm{O}_{2}$ transferred to the liquid is small relative to molar flow rate of $\mathrm{O}_{2}$ in the aeration gas. a. Determine the dissolved oxygen concentration after a 5.0 min aeration time. b. Determine the time required to bring the dissolved oxygen concentration up to a concentration of 0.20 mmole/L (1 mmole/L = 1 gmole/m$^{3}$).

Using R and S descriptors, write all the possible combinations for a molecule with three chirality centers.

The table shows a familiar pattern from geometry. (a) Determine what the variables x and y represent. Then write a function rule that relates y to x. (b) Is the function a linear function? Explain your reasoning.

$$\begin{matrix} \text{x} & \text{1} & \text{2} & \text{3} & \text{4} & \text{5}\\ \text{y} & \text{2} & \text{4} & \text{6} & \text{8} & \text{10}\\ \end{matrix}$$

Imelda operates a home business selling two computer games: "Space Raiders" and "Green Giants." These games are installed for Imelda by three friends, Nicolas, Harvey, and Karl, each of whom must do some of the work on the installation of each game. The time that each must spend on each game is given in the following table:

$$\begin{array}{cccc} {} & {\text { Nicolas }} & {\text { Harvey }} & {\text { Karl }} \\ \hline \text { Space Raiders } & {30 \mathrm{min}} & {20 \mathrm{min}} & {10 \mathrm{min}} \\ \text { Green Giants } & {10 \mathrm{min}} & {10 \mathrm{min}} & {50 \mathrm{min}} \end{array}$$

Imelda's friends have other work to do, but they each find that each week they can spend at most 300 minutes working on Imelda's games. Imelda makes a profit of $10 on each sale of Space Raiders and$15 on each sale of Green Giants. How many of each game should Imelda try to sell each week to maximize profit, and what is this maximum profit?

Differentiate the functions. $f(x)=x^{-6}$

An object moves in rectilinear motion so that at time $t \geq 0$ seconds, its position from the origin is $s(t)=\sin e^{t}$, in feet. (a) Find the velocity v and acceleration a of the object at any time t. (b) At what time does the object first have zero velocity? (c) What is the acceleration of the object at the time t found in (b)?

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. $f(x)=\sqrt{x}+2$

Do not use the simplex method. Minimize $Z=2 x+3 y$ subject to

$$\begin{aligned} x+y & \leq 5 \\ 2 x+5 y & \leq 10 \\ 5 x+8 y & \geq 20 \\ x, y & \geq 0 \end{aligned}$$

Evaluate the integral. $\int \tan x \ln (\cos x) d x$

In a discussion of the delivered price of a good from a mill to a customer, DeCanio$^{3}$ arrives at and solves the two quadratic equations $(2 n-1) v^{2}-2 n v+1=0$ and $n v^{2}-(2 n+1) v+1=0$ where $n \geq 1.$ (a) Solve the first equation for v. (b) Solve the second equation for v if v < 1.

Find each indefinite integral. $\int \sin x \cos ^{2} x d x$

Use either the shell method or the disk/washer method to find the volume of the solid of revolution generated by revolving the region bounded by the graphs of the given equations about the indicated axis. $y=x^{2 / 3}+x^{1 / 3}$, the x-axis, x=1, x=8; about the y-axis

Find $f^{-1}$ and its domain. $f(x)=\sqrt{3-e^{2 x}}$

Solve each equation for x. $\ln \left(x^{2}-1\right)=3$