Suppose that an honest die is rolled n=180 times. Let the random variable X represent the number of times the number 6 is rolled. (a) Find the mean and standard deviation for the distribution of X. (Hint: Use the dishonest-coin principle with p=1/6 to find

$$\mu$$

and

$$\sigma$$

.) (b) Find the probability that a 6 will be rolled more than 40 times. (c) Find the probability that a 6 will be rolled somewhere between 30 and 35 times.

Rayleigh’s criterion is used to determine when two objects are barely resolved by a lens of diameter d. The angular separation must be greater than $\theta_{R}$ where $\theta_{R} =1.22 \lambda /d$ . In order to resolve two objects 4000 nm apart at a distance of 20 cm with a lens of diameter 5 cm, what energy (a) photons or (b) electrons should be used? Is this consistent with the uncertainty principle?

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

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

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?

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.

Why is the following situation impossible? An air rifle is used to shoot 1.00-g particles at a speed of $v_{x}=100 \mathrm{m} / \mathrm{s} .$ The rifle's barrel has a diameter of 2.00 mm. The rifle is mounted on a perfectly rigid support so that it is fired in exactly the same way each time. Because of the uncertainty principle, however, after many firings, the diameter of the spray of pellets on a paper target is 1.00 cm.

(a) Derive an expression for the angle between the vector representing angular momentum l with z-component $m_{1}=+l$ (that is, its maximum value) and the z-axis. What is this angle for l = 1 and for l = 5? (b) Derive an expression for the angle between the vector representing angular momentum l with z-component $m_{1}=+l$ and the z-axis. What value does this angle take in the limit that l becomes very large? Interpret your result in the light of the correspondence principle.

Refer to the figure and use the principle of superposition to Find the component of the current $i$ through $R_{3}$ that is due to $V_{S 2}$.

$$\begin{aligned} & V_{S 1}=V_{S 2}=450 \mathrm{~V} \\ & R_{1}=7 \Omega \quad R_{2}=5 \Omega \\ & R_{3}=10 \Omega \quad R_{4}=R_{5}=1 \Omega \end{aligned}$$

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?

(a) Which of the compounds shown in mentioned figure can in principle be resolved into enantiomers? Explain why or why not. (b) Omeprazole can be separated into two enantiomers that do not interconvert at room temperature. What atom is the asymmetric center in omeprazole? (c) Esomeprazole, the $S$ enantiomer of omeprazole, is a drug used to control acid reflux. Redraw the structure of omeprazole in part (b) to show it as the $S$ enantiomer. (Hint: An unshared electron pair has lower priority than H.)

Use the uncertainty principle to estimate the binding energy of the $\mathrm{H}_2$ molecule by calculating the difference in kinetic energy of the electrons between when they are in separate atoms and when they are in the molecule. Take $\Delta x$ for the electrons in the separated atoms to be the radius of the first Bohr orbit, $0.053 \mathrm{~nm}$, and for the molecule take $\Delta x$ to be the separation of the nuclei, $0.074 \mathrm{~nm}$. [Hint: Let $\left.\Delta p \approx \Delta p_x.\right]$

You have been involved with a school committee trying to raise money through bake sales. At the last bake sale, you sold only half your goods. What could you suggest to your committee that might ensure more quantities are sold? Work with a partner to make a plan for the next bake sale. Make sure you refer to the Law of Demand and the principle of diminishing marginal utility as you prepare your suggestions.

Object A has mass m$_A$ = 10 kg and initial momentum p$_{A,i}$ = (21, -7, 0) kg·m/s, just before it strikes object B, which has mass m$_B$= 14 kg. Just before the collision object B has initial momentum p$_{B,i}$= (3, 6, 0) kg·m/s. Therefore, what does the Momentum Principle predict that the total final momentum of the system will be, just after the collision?

Use the duck principle to show that each equation is true. a. $\sqrt[4]{125} \cdot \sqrt[4]{5} = 5$ b. $\sqrt[3]{18} \cdot \sqrt[3]{12} = 6$ c. $\dfrac{\sqrt[5]{128}}{\sqrt[5]{16}} = \sqrt[5]{8}$ d. $\sqrt[4]{100} = \sqrt{10}$ e. $\sqrt[3]{4} \cdot \sqrt[6]{3} = \sqrt[6]{48}$