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The solution provides step-by-step calculations for relative and percent frequency distributions, bar graphs, pie charts, frequency distributions, cumulative frequency distributions, histograms, frequency polygons, and measures of central tendency (mean, median, mode) and dispersion (variance, standard deviation) for sets of data related to soft drink purchases and audit times.

The provided text details the process of performing LU decomposition on a 4x4 matrix, including row swaps and elementary row operations to achieve an upper triangular matrix (U) and a lower triangular matrix (L). It also calculates the corresponding diagonal matrix (D) and permutation matrix (P) to complete the decomposition. The solution demonstrates the Gaussian elimination method to find the inverse of a 3x3 matrix.

The equation of the tangent line to the circle (x-3)^2 + (y-2)^2 = 25 at the point (-1, 5) is 4x - 3y + 19 = 0.

The solution finds the symmetric difference between two sets derived from inverse functions, demonstrating the application of set theory and function operations. It also proves a set identity involving unions and intersections of sets, and calculates derivatives of composite functions using the chain rule. Finally, it calculates probabilities of events involving coin tosses and dice rolls, and determines the probability of having 53 Sundays or Mondays in a leap year.

The determinant is evaluated by splitting it into two parts, factoring out terms, and using the Vandermonde determinant formula. The final result is expressed as a product of factors involving x, y, and z, and a term (1-xyz).

The area of the unknown rectangle (A4) is calculated to be 6 square meters by using the given areas of the other rectangles and the relationships between their dimensions.

The system of equations 6^x + 6^y = 42 and x + y = 3 has two solutions: (x, y) = (2, 1) and (x, y) = (1, 2).

The solution calculates confidence intervals for the variance and mean of bolt diameters, demonstrating how reducing the confidence level results in a narrower confidence interval for the mean.

The solution calculates 95% confidence intervals for a population mean using both the t-distribution (when population variance is unknown) and the z-distribution (when population variance is known). It also discusses how increasing the confidence level results in a wider confidence interval.

The 95% confidence interval for the population mean is [66.04, 69.96], calculated using a sample mean of 68, a known variance of 22, and a sample size of 22. The minimum sample size needed to achieve a maximum confidence interval width of 1 is 338.

The 95% confidence interval for the population mean is approximately [13.984, 14.416]

The provided text details step-by-step calculations for confidence intervals of means and variances, along with sample size determination, under different scenarios (known and unknown population standard deviation). The solutions involve using z-scores, t-scores, and chi-squared values to construct intervals for population parameters.

The 99% confidence interval for the population mean length of steel sheets is [35.71, 38.29] cm.

The Laplace transform of the piecewise function f(t) is calculated using Heaviside step functions and the time-shifting property of Laplace transforms. The result is expressed in terms of the exponential function and the reciprocal of s squared.

The provided text describes functions, relations, domains, ranges, and inverse functions. It includes examples of relations that are not functions, and explains how to determine the domain and range of a function, including a quadratic function and a rational function. It also demonstrates how to find the inverse of a function.

The solution to the system of equations 5m + n = 26 and m - n = 4 is m = 5 and n = 1. Calculating m^n results in 5^1 = 5.

To solve for x in the equation 2x + 5 = 13, subtract 5 from both sides, then divide both sides by 2 to get x = 4.

The expression evaluates to 0 after calculating the numerator and denominator separately and then dividing them.

The sum of three angles around a point, 2x, x, and x, equals 360 degrees, leading to the solution x = 90 degrees.

To find the unknown interior angle x, the exterior angle theorem was used, setting up and solving an equation to find x = 40 degrees.

The provided text contains a series of math problems, including algebraic expressions, exponents, fractions, and polynomial operations. The solutions demonstrate various techniques for simplifying and manipulating these expressions, such as factoring, expanding, and applying the rules of exponents and fractions.

The function f(x,y) = -2x^2 + 3xy^2 - 3y^2 + 4 has a local maximum value of 4 at (0,0).

The solution determines the Fourier coefficients for a periodic square pulse with a width of 3 and period of 12, finding the values of α, β, and λ in the given formula for ak.

The derivative of tan(θ) with respect to θ is sec^2(θ), derived using right-angled triangles and the definition of the derivative.

The provided examples demonstrate the continuity of functions at specific points. Polynomials are continuous everywhere, functions with undefined points are not continuous at those points, and absolute value functions are continuous everywhere, including at points where the expression inside the absolute value is zero.

The double integral, using polar coordinates and integration by parts, evaluates to π(e^2 + 1).

Roulette outcomes are random and independent, making future predictions impossible based on past results. Each number has an equal probability of appearing.

The solution to the system of linear equations is x = 1, y = -2, z = 3, and w = 2.

Given a^2 - 4 = 0, the possible values for a are 2 and -2. Substituting these values into a^3 - 4, we find that a^3 - 4 can be either 4 or -12.

The solution attempts to derive a reduction formula for the integral \(\int_0^\infty \frac{x}{\cosh^n(x)} dx\) using integration by parts and Feynman's trick, but ultimately finds that expressing the integral in terms of hypergeometric functions is a more suitable approach.