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Find the maximum value of an algebraic expression given a constraint on the sum of three positive variables. The solution involves applying algebraic manipulations and inequalities to determine the upper bound of the expression.

Solve the second-order linear differential equation y'' - y = 0 using the Method of Undetermined Power Series Coefficients, resulting in a solution involving hyperbolic functions.

Solve the algebraic equation A + AA + AAA = 738 to find the value of the digit A. The solution involves simplifying the equation and solving for A.

Solve the algebraic equation (a+2)(a+3)(a+4)(a+5) / (a-2)(a-3)(a-4)(a-5) = 1 to find the value of a. The solution involves simplifying the equation and solving for a.

Solve the division problem involving fractions: (55 / 5) / (5 / 55). The solution involves simplifying the numerator and denominator, then dividing the resulting fractions.

Solve the linear equation 2x + 5 = 12 to find the value of x. The solution involves isolating x by subtracting and dividing.

Solve for X in the equation X = √(100² - 96²). This problem involves simplifying the expression using the difference of squares formula and calculating the square root.

The equation (X-1)^2 = X-1 has two solutions, X = 1 and X = 2.

Solve the given system of linear equations by using the elimination method to find the values of x and y.

Solve the equation involving division of fractions and find the value of X.

The solutions to the equation \sqrt{5x^2-6x+8} - \sqrt{5x^2-6x-7} = 1 are x = 4 and x = -14/5.

The provided text contains a detailed solution to various integration problems, including trigonometric substitutions, partial fraction decomposition, and integration by parts. It also demonstrates the use of properties of definite integrals to simplify calculations.

The area of a square with side length 3 is 9.

The solution calculates quartiles, deciles, percentiles, and the coefficient of skewness for a dataset of 21 values. It determines the dataset is negatively skewed.

The solution calculates the range, arithmetic mean, population variance, and standard deviation of the annual incomes of five vice presidents.

The mixed strategy Nash equilibria are (Pass, Pass) and (Two, Two). For (Pass, Pass), the expected utility for each player is 1. For (Two, Two), the expected utility for each player is 0.

The solution calculates quartiles, deciles, percentiles, and the coefficient of skewness for a dataset of 21 values. It demonstrates the relationships between these measures and concludes that the distribution is slightly negatively skewed.

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.