answer:a. The energy of the electron in an {eq}He^+ {/eq} ion ({eq}Z=2 {/eq}) is given by: E_n = frac{Z^2(-13.6 eV)}{n^2} For the third excited state ({eq}n=4 {/eq}), the energy is: E_3 = frac{(2)^2(-13.6 eV)}{4^2} = -3.4 rm eV b. Using the Rydberg formula, we find the wavelengths ({eq}lambda {/eq}) for transitions from {eq}n=4 {/eq} to: 1. {eq}n=3 {/eq}: frac{1}{lambda_{4 to 3}} = 2^2R_Hleft(frac{1}{3^2}-frac{1}{4^2}right) = 468.651626 text{nm} 2. {eq}n=2 {/eq}: frac{1}{lambda_{4 to 2}} = 2^2R_Hleft(frac{1}{2^2}-frac{1}{4^2}right) = 25.6293858 text{nm} 3. {eq}n=1 {/eq} (ground state): frac{1}{lambda_{4 to 1}} = 2^2R_Hleft(frac{1}{1^2}-frac{1}{4^2}right) = 24.30045468 text{nm} c. The ionization energy is the energy needed to remove the electron from its current state. For the third excited state ({eq}n=3 {/eq}), this is: E_{ionization} = -E_3 = 6.04 rm eV d. To ionize the electron from the third excited state, it must transition from {eq}n=4 {/eq} to {eq}n=infty {/eq}. The energy of the photon is: frac{hc}{lambda_4} = frac{(2)^2(-13.6 eV)}{(4)^2} = 3.4 rm eV lambda_4 = frac{hc}{3.4 rm eV} = 364.50682024 text{nm} e. Using Bohr's radius formula for an {eq}He^+ {/eq} ion, we find the radius of the third excited state: r_n = frac{n^2h^2}{Z^2m_ealpha^2} where {eq}n=4 {/eq}, {eq}h=6.626 times 10^{-34} text{J} cdot text{s} {/eq}, {eq}Z=2 {/eq}, and {eq}alpha {/eq} is the fine-structure constant. r_4 = frac{(4)^2(6.626 times 10^{-34} text{J} cdot text{s})^2}{(2)^2(9.109 times 10^{-31} text{kg})(7.297 times 10^{-3} text{m})^2} = 4.2334 times 10^{-10} text{m} f. The electrostatic potential energy is: U = -frac{k_ee^2}{r} where {eq}r = r_4 {/eq} from part (e). U = -frac{left(dfrac{1}{4pi epsilon_0}right)(1.602 times 10^{-19} text{C})^2}{4.2334 times 10^{-10} text{m}} = 3.4014 rm eV Note that the potential energy is negative, indicating that the electron is bound to the nucleus.

answer:Given Data: The given equation is, {eq}{{sin }^{2}}x-4cos x-4=0{/eq}, where , {eq}0{}^circ le x<360{}^circ {/eq}. Using the identity, {eq}{{sin }^{2}}x=1-{{cos }^{2}}x{/eq}: {eq}begin{align} {{sin }^{2}}x-4cos x-4&=0 1-{{cos }^{2}}x-4cos x-4&=0 {{cos }^{2}}x+4cos x+3&=0 left( cos x+3 right)left( cos x+1 right) &=0 end{align}{/eq} Therefore, either {eq}cos x=-3,,text{or},,cos x=-1{/eq}. Since cosine function lies between -1 and 1, {eq}cos x=-3{/eq} has no solution. If {eq}cos x=-1{/eq}, then {eq}x={{cos }^{-1}}left( -1 right)Rightarrow x=180{}^circ {/eq}. Hence the solution of the equation {eq}{{sin }^{2}}x-4cos x-4=0{/eq} is {eq}180{}^circ {/eq}.

answer:The Clean Air Act and the Endangered Species Act

answer:Covariance is given by the formula: Covariance = {eq}displaystyle frac { sum_{i=1}^{n} (R_{i,EL} - Average return_{EL})(R_{i,LC} - Average return_{LC}) } {n-1} {/eq} where: * R_i,EL is the return of Estee Lauder in year i * Average return_EL is the average return of Estee Lauder * R_i,LC is the return of Lowe's Companies in year i * Average return_LC is the average return of Lowe's Companies Substituting the given values: Covariance = {eq}displaystyle frac { (23.4 - 9.62)(-6 - 8.66) + (-26 - 9.62)(16.1 - 8.66) + (17.6 - 9.62)(4.2 - 8.66) + (49.9 - 9.62)(48 - 8.66) + (-16.8 - 9.62)(-19 - 8.66) } {5-1} {/eq} = 576.76 Correlation coefficient is given by the formula: Correlation coefficient = {eq}displaystyle frac {Covariance} {Standard deviation_{EL} * Standard deviation_{LC}} {/eq} Substituting the given values: Correlation coefficient = {eq}displaystyle frac {576.76} {31.00 * 25.51} {/eq} = 0.727 Interpretation: The covariance of 576.76 indicates a positive relationship between the returns of Estee Lauder and Lowe's Companies. The correlation coefficient of 0.727 suggests a strong positive correlation, implying that the returns of these two stocks tend to move in the same direction.