Write the combined gas law equation. ________ Pressure is inversely proportional to ______ and directly proportional to _______. Volume also is _________ to temperature.

Write the mathematical expression for the combined gas law.

The pressure of a gas stored in a refrigerated container is 4.0 atm at $22.0^{\circ} \mathrm{C}$. Determine the gas pressure in the tank if the temperature is lowered to $0.0^{\circ} \mathrm{C}$.

1. Analyze the Problem Known: $P_{1}$ = 4.0 atm $T_{1}$ = ____ $T_{2}$ = ____ Unknown: $P_{2}$ = ? ____ Use Gay-Lussac's law and the known values for $T_{1}$, $V_{1},$ and $T_{2}$ to solve for $V_{2}$.
2. Solve for the Unknown Convert the $T_{1}$ and $T_{2}$ Celsius figures to kelvin.

$$\begin{array}{l} T_{1}= \text{\_\_\_}+22.0^{\circ} \mathrm{C}= \text{\_\_\_}{\mathrm{K}} \\ T_{2}=273+ \text{\_\_\_}^{\circ} \mathrm{C}=\text{\_\_\_}{\mathrm{K}} \end{array}$$

Write the equation for Gay-Lussac's law. To solve for $P_{2}$, multiply both sides by $T_{2}$. $P_{2}$ = ____ Substitute known values. $P_{2}$ = ____ Solve for $P_{2}$. $P_{2}$ = 3.7 atm 3. Evaluate the Answer The temperature ______ and the pressure _____.

Helium gas in a balloon is compressed from 4.0 L to 2.5 L at constant temperature. The gas's pressure at 4.0 L is 210 kPa. Determine the pressure at 2.5 L.

1. Analyze the Problem Known: $V_{1}$ = _____ $V_{2}$ = _____ $P_{1}$ = _____ Unknown: $P_{2}$ = _____ Use the equation for Boyle's law to solve for $P_{2}$.
2. Solve for the Unknown Write the equation for Boyle's law: __________ To solve for $P_{2}$, divide both sides by $V_{2}$. $P_{2}$ = _________ Substitute the known values. $P_{2}$ = _________ Solve for $P_{2}$. $P_{2}$ = _________
3. Evaluate the Answer When the volume is _____, the pressure is _____. The answer is in ___, a unit of pressure.

List the left and right cosets of the subgroups in each of the following. (a) $\langle 8\rangle$ in $\mathbb{Z}_{24}$, (b) $\langle 3\rangle$ in $U(8)$, (c) $3 \mathbb{Z}$ in $\mathbb{Z}$, (d) $A_{4}$ in $S_{4}$, (e) $A_{n}$ in $S_{n}$, (f) $D_{4}$ in $S_{4}$, (g) $\mathbb{T}$ in $\mathbb{C}^{*}$, (h) $H=\{(1),(123),(132)\}$ in $S_{4}$.

A car rental agency charges a flat fee of $30 to rent a car, and$21 for each day the car is rented. The linear function c=30+21d can be used to calculate the customer's cost (c) based on the number of days (d) the car is rented. What is the maximum number of days Lakesha can rent a car if she has only $140 to spend?

For the following exercises, find the inverse of the functions. $f(x)=4-x^{3}$

The table gives salaries and number of workers at a manufacturer of auto parts.

$$\begin{array}{|l|c|c|c|c|c|} \hline \text { Job } & \text { President } & \begin{array}{c} \text { Group } \\ \text { Manager } \end{array} & \begin{array}{c} \text { Line } \\ \text { Manager } \end{array} & \text { Machinist } & \text { Clerk } \\ \hline \begin{array}{l} \text { Number } \\ \text { Salary Range } \end{array} & \begin{array}{c} 1 \\ \mathbf{\$ 8 1 - 9 5 , 0 0 0} \end{array} & \begin{array}{c} \mathbf{2} \\ \mathbf{\$ 6 6 - 8 0 , 0 0 0} \end{array} & \begin{array}{c} \mathbf{5} \\ \mathbf{\$ 5 1 - 6 5 , 0 0 0} \end{array} & \mathbf{\$ 3 5 - 5 0 , 0 0 0} & \mathbf{2 5} & \mathbf{4} \\ \hline \end{array}$$

In a lavor dispute, which measures of central tendency might the president use to show that the worker's wages were already high enough? Why?

For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question. A right circular cylinder is to have a volume of 40 cubic inches. It costs 4 cents/square inch to construct the top and bottom and 1 cent/square inch to construct the rest of the cylinder. Find the radius to yield minimum cost. Let x = radius.

Use transformations to describe how the graph of tie function is related to a basic inverse trigonometric graph. Slate the omain and range. $g(x)=3 \arccos (x / 2)$

For the following exercises, rewrite the quadratic functions in standard form and give the vertex. $f(x)=2 x^{2}-6 x$

Consider the graph of an ellipse C, oriented counterclockwise. Graphs of the vector fields $\mathbf{F}_{1}, \mathbf{F}_{2}, \mathbf{F}_{3}$, and $\mathbf{F}_{4}$ along the curve C are given. Along C, $\mathbf{F}_{1}$ and $\mathbf{F}_{4}$ are tangent to C, and $\mathbf{F}_{2}$ and $\mathbf{F}_{3}$ are normal to C. Determine whether the following integrals are positive, negative, or equal to 0. a. $\int_{C} \mathbf{F}_{1} \cdot \mathbf{T} d s$ b. $\int_{C} \mathbf{F}_{2} \cdot \mathbf{T} d s$ c. $\int_{\mathbf{C}} \mathbf{F}_{\mathbf{3}} \cdot \mathbf{T} \mathbf{d} \mathbf{s}$ d. $\int_{\boldsymbol{C}} \mathbf{F}_{{4}} \cdot \mathbf{T} d \boldsymbol{s}$ e.$\int_{c} \mathbf{F}_{1} \cdot \mathbf{n} d s$ f. $\int_{\boldsymbol{C}} \mathbf{F}_{2} \cdot \mathbf{n} d \boldsymbol{s}$ g. $\int_{C} \mathbf{F}_{3} \cdot \mathbf{n} d s$ h. $\int_{C} \mathbf{F}_{4} \cdot \mathbf{n} d s$

Solve each equation. $b-5=-41$

The graph of the equation $\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=0$ is considered to be a degenerate hyperbola. Describe the graph. How is it like a full-fledged hyperbola, and how is it different?