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March 16, 2025

Statistics

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.

March 14, 2025

Statistics

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.

March 9, 2025

Linear-Algebra

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.

March 2, 2025

Geometry

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.

February 27, 2025

Pre-Algebra

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.

February 23, 2025

Pre-Algebra

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).

February 19, 2025

Geometry

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.

February 19, 2025

Pre-Algebra

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).

February 11, 2025

Statistics

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.

February 11, 2025

Statistics

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.