Underline any verbs that are used incorrectly. In the space provided, write the correct form (or $C$ if correct) and the number of the $\mathrm{G} / \mathrm{M}$ principle illustrated. When you finish, compare your responses with those provided at the bottom of this page. If your responses differ, study carefully the principles in parentheses.

Everyone except the temporary workers employed during the past year has become eligible for health benefits.

The following exercise is a problem that may require permutations, combinations, or the multiplication principle.

According to the Baskin-Robbins Web site, there are 21 "classic flavors" of ice cream.

(a) How many different double-scoop cones can be made if order does not matter (for example, putting chocolate on top of vanilla is equivalent to putting vanilla on top of chocolate)?

(b) How many different triple-scoop cones can be made if order does matter?

Solve each minimum problem using the duality principle. Minimize

$$C = x_1 +4x_2 + 2x_3 +4x_4$$

subject to the constraints 2

$$\left\{ \begin{array}{ccccccc} x_1 & & +x_3 & \geq 1 \\ & x_2 & & +x_4 \geq1 \\ - x_1 &-x_2 &-x_3 &-x_4 \geq -3 \\ x_1 \geq 0, & x_2 \geq 0, & x_3 \geq 0 & x_4 \geq 0 \end{array} \right.$$

Applying Le Chatelier's principle, in which direction will the equilibrium shift, (if at all)?

$$\mathrm{CH}_4(\mathrm{~g})+2 \mathrm{~O}_2(\mathrm{~g}) \rightleftharpoons \mathrm{CO}_2(\mathrm{~g})+2 \mathrm{~H}_2 \mathrm{O}(\mathrm{g})+802.3 \mathrm{~kJ}$$

(a) if the temperature is increased
(b) if a catalyst is added
(c) if $\mathrm{CH}_4$ is added
(d) if the volume of the reaction vessel is decreased

When 2.00 mo! of $SO_2Cl_2$ is placed in a 2.00-L fla sk at 303 K, 56% of the $SO_2Cl_2$ decomposes to $SO_2$ and $Cl_2:$ $\mathrm { SO } _ { 2 } \mathrm { Cl } _ { 2 } ( g ) \rightleftharpoons \mathrm { SO } _ { 2 } ( g ) + \mathrm { Cl } _ { 2 } ( g )$ According to Le Chatelier's principle, would the percent of $SO_2Cl_2$ that decomposes increase, decrease or stay the same if the mixture were transferred to a 15.00-L vessel?

Solve each minimum problem using the duality principle. Minimize

$$C = x_1 +2x_2 +x_3$$

subject to the constraints 2

$$\left\{ \begin{array}{rcl} x_1 & -3x_2 & +4x_3 \geq 12 \\ 3x_1 & +x_2 & +2x_3 \geq 10 \\ x_1 & -x_2 & -x_3 \geq -8 \\ x_1 \geq 0, & x_2 \geq 0, & x_3 \geq 0 \end{array} \right.$$

Suppose $f$ is differentiable at points on a closed path $\gamma$ and at all points in the region G enclosed by $\gamma,$ except possibly at a finite number of poles of $f$ in G. Let Z be the number of zeros of $f$ in G, and P the number of poles of $f$ in G, with each zero and pole counted as many times as its multiplicity. Show that $\frac{1}{2 \pi i} \oint_{\gamma} \frac{f^{\prime}(z)}{f(z)} d z=Z-P.$ This formula is known as the argument principle.

Show how a single block of glass can be used to turn a p-polarized beam of light through $180^{\circ},$ with the light suffering (in principle) zero reflective loss. The light is incident from air, and the returning beam (also in air) may be displaced sideways from the incident beam. Specify all pertinent angles and use n = 1.45 for glass. More than one design is possible here.

Third-Order Examples: For each of the nonhomogeneous linear DEs in $(a)$ Ver(fy that the given $y_{1}, y_{2}, y_{3}$ satisfy the corresponding homogeneous equation. $(b)$Use the Superposition Principle, with appropriate coefficients, to state the general solution $y_{1}(t)$ to the corresponding homogeneous equation. $(c)$ Verify that the given $y_{p}(t)$ is a particular solution to the given nonhomogeneous s DE. $(d)$ Use the Nonhomogeneous Principle to write the general solution $y(t)$ to the nonhomogeneous DE. $(e)$ Solve the NP consisting of the nonhomogeneous DE and the given initial conditions

$$\begin{array}{l}{y^{\prime \prime}-y^{\prime \prime}-y^{\prime}+y=2 t-1+3 e^{2 t}} \\ {y_{1}(t)=e^{t}, y_{2}(t)=t e^{t}, y_{3}(t)=e^{-t}} \\ {y_{p}(t)=2 t+1+e^{2 t}} \\ {y(0)=4, y^{\prime}(0)=3, y^{\prime \prime}(0)=4}\end{array}$$

use Archimedes’ Principle, which states that when a sphere of radius r with density

$$d_s$$

is placed in a liquid of density

$$d_L=62.5lb/ft^3$$

,it will sink to a depth h where .

$$\dfrac{π}{3}(3rh^2-h^3)d_L=\dfrac{4}{3}πr^3d_s$$

Find an approximate value for h if:

$$r=5 ft$$

and

$$d_s=20lb/ft^3$$

In a population of rabbits, f (C$_{1}$) = $0.70$ and f (C$_{2}$) = $0.30$. The alleles exhibit an incomplete dominance relationship in which C$_{1}$C$_{1}$ produces black rabbits, C$_{1}$C$_{2}$ tan-colored rabbits, and C$_{2}$C$_{2}$ rabbits with white fur. If the assumptions of the Hardy-Weinberg principle apply to the rabbit population, what are the expected frequencies of black, tan, and white rabbits?

What is the expected result when you roll a single die? Specify 40 rolls, and then click on "roll the dice." After the last roll, click on "show mean." Does your expected sum agree with the green mean? What happens to the plotted points as the number of rolls increases? If the picture is not clear, click on "roll the dice" for a new total of 80 rolls. Describe what you see in a sentence. What is the name for this principle?

Magnificent Blooms is a florist specializing in floral arrangements for weddings, graduations, and other events. The firm has a fixed cost associated with space and equipment of $100 per day. Each worker is paid$50 per day. The daily production function for Magnificent Blooms is shown in the accompanying table.

$$\begin{array}{cc} \begin{array}{c} \text { Quantity of labor } \\ \text { (workers) } \end{array} & \begin{array}{c} \text { Quantity of floral } \\ \text { arrangements } \end{array} \\ \hline 0 & 0 \\ \hline 1 & 5 \\ \hline 2 & 9 \\ \hline 3 & 12 \\ \hline 4 & 14 \\ \hline 5 & 15 \\ \hline \end{array}$$

Calculate the marginal product of each worker. What principle explains why the marginal product per worker declines as the number of workers employed increases?

Photodissociation of water

$$\mathrm{H}_2 \mathrm{O}(l)+h v \longrightarrow \mathrm{H}_2(g)+\frac{1}{2} \mathrm{O}_2(g)$$

has been suggested as a source of hydrogen. The energy for the reaction is $285.8 \mathrm{~kJ}$ per mole of water decomposed. Calculate the maximum wavelength (in $\mathrm{nm}$ ) that would provide the necessary energy. In principle, is it feasible to use sunlight as a source of energy for this process?