answer:When the first ball is drawn, there is a {eq}frac{5}{12} {/eq} chance that it is white, a {eq}frac{3}{12} {/eq} chance that it is red, and a {eq}frac{4}{12} {/eq} chance that it is blue. Since there is no replacement, for the second ball, there is a {eq}frac{4}{11} {/eq} chance that it is white, a {eq}frac{3}{11} {/eq} chance that it is red, and a {eq}frac{4}{11} {/eq} chance that it is blue. Continuing in this manner, for the third ball, there is a {eq}frac{3}{10} {/eq} chance that it is white, a {eq}frac{2}{10} {/eq} chance that it is red, and a {eq}frac{4}{10} {/eq} chance that it is blue. Finally, for the fourth ball, there is a {eq}frac{2}{9} {/eq} chance that it is white, a {eq}frac{1}{9} {/eq} chance that it is red, and a {eq}frac{3}{9} {/eq} chance that it is blue. To find the total probability, multiply the probabilities of each drawing together: {eq}begin{align*} text{ Probability } &= frac{5}{12}cdot frac{4}{11}cdot frac{3}{10}cdot frac{2}{9} &+ frac{5}{12}cdot frac{4}{11}cdot frac{2}{10}cdot frac{3}{9} &+ frac{5}{12}cdot frac{3}{11}cdot frac{4}{10}cdot frac{2}{9} &+ frac{5}{12}cdot frac{3}{11}cdot frac{2}{10}cdot frac{4}{9} &+ frac{3}{12}cdot frac{5}{11}cdot frac{4}{10}cdot frac{2}{9} &+ frac{3}{12}cdot frac{5}{11}cdot frac{2}{10}cdot frac{4}{9} &+ frac{3}{12}cdot frac{4}{11}cdot frac{5}{10}cdot frac{2}{9} &+ frac{3}{12}cdot frac{4}{11}cdot frac{2}{10}cdot frac{5}{9} &+ frac{4}{12}cdot frac{5}{11}cdot frac{3}{10}cdot frac{2}{9} &+ frac{4}{12}cdot frac{5}{11}cdot frac{2}{10}cdot frac{3}{9} &+ frac{4}{12}cdot frac{3}{11}cdot frac{5}{10}cdot frac{2}{9} &+ frac{4}{12}cdot frac{3}{11}cdot frac{2}{10}cdot frac{5}{9} &= frac{8}{99} end{align*} {/eq} Therefore, the probability of selecting 1 ball of each color is {eq}frac{8}{99} {/eq}.

answer:(A) The given differential equation is {eq}displaystyle x'' + x = 5t e^{2t} {/eq} The characteristic equation of this ODE is {eq}displaystyle m^2 + 1 = 0 m^2 = -1 m = pm i {/eq} The homogeneous solution is {eq}displaystyle x_h(t) = C_1 cos t + C_2 sin t {/eq} For the particular solution, we consider {eq}displaystyle x_p(t) = At e^{2t} + Be^{2t} {/eq} Taking derivatives: {eq}displaystyle x_p'(t) = 2At e^{2t} + (A + 2B)e^{2t} {/eq} {eq}displaystyle x_p''(t) = 4At e^{2t} + (4A + 4B)e^{2t} {/eq} Substituting into the ODE: {eq}displaystyle x_p'' + x_p = 5t e^{2t} 4At e^{2t} + (4A + 4B)e^{2t} + At e^{2t} + Be^{2t} = 5t e^{2t} 5At e^{2t} + (4A + 5B)e^{2t} = 5t e^{2t} {/eq} Comparing coefficients: {eq}displaystyle A = 1, quad 4A + 5B = 0 A = 1, quad B = -frac{4}{5} {/eq} The particular solution is {eq}displaystyle x_p(t) = t e^{2t} - frac{4}{5} e^{2t} {/eq} Hence, the general solution is {eq}boxed{displaystyle x(t) = x_h(t) + x_p(t) x(t) = C_1 cos t + C_2 sin t + t e^{2t} - frac{4}{5} e^{2t}} {/eq} (B) The differential equation is {eq}displaystyle x'' + x = 5te^{2t} {/eq} If {eq}y(t) = At e^{2t} + Be^{2t} {/eq} is a particular solution with {eq}y(0) = 1, y'(0) = 2, {/eq} we get {eq}displaystyle B = 1, quad A + 2B = 2 {/eq} Solving these equations gives {eq}displaystyle A = 0, quad B = 1 {/eq} So, the particular solution is {eq}displaystyle y(t) = e^{2t} {/eq} The general solution is {eq}displaystyle x(t) = x_h(t) + y(t) x(t) = C_1 cos t + C_2 sin t + e^{2t} x'(t) = -C_1 sin t + C_2 cos t + 2e^{2t} {/eq} Applying initial conditions {eq}x(0) = 3, x'(0) = 5, {/eq} we obtain {eq}displaystyle C_1 = 2, quad C_2 = 3 {/eq} Thus, the solution is {eq}boxed{displaystyle x(t) = 2cos t + 3sin t + e^{2t}} {/eq}

answer:Forensic science is a multidisciplinary field that integrates scientific methods and techniques to gather and analyze evidence for use in civil and criminal legal proceedings. Experts in forensic science examine and interpret various forms of evidence, such as DNA samples, fingerprints, and chemical substances, to assist investigators in identifying perpetrators, establishing timelines, and reconstructing events at crime scenes. Advanced technologies in forensic science enable the reexamination of old evidence, potentially leading to the resolution of previously unsolved cases and ensuring the accurate identification of individuals involved.

answer:The perimeter of the quadrilateral is 2.59 units. It is a convex quadrilateral. The interior angles are estimated to be {0.91, 2.8, 1.32, 1.25} radians. The area of the quadrilateral is 0.34 square units.