Uncovering the Area Relationship between a Histogram and a Right Trapezoid

Abstract

English translation of the Thai abstract on page ก (PDF page 3) of the original project report. This mathematics project has two objectives: (1) to investigate the area relationship between a histogram and a right trapezoid using the area of a right trapezoid and quadratic equations in one variable; and (2) to check the resulting formula using GSP. The authors consider data whose successive frequencies increase by an equal amount, calculated as a fixed percentage of the first value. They report that representing these data as a histogram leads to an area relationship with a right trapezoid. This observation motivates an investigation of the relevant variables and the derivation of a formula using the area formula for a right trapezoid and a quadratic equation in one variable. The authors report that checks performed using GSP produced the expected results. On this basis, they conclude that their formula is mathematically correct and can be applied in practice. The relationships given in the abstract are reproduced below. Let y₁ denote the first data value, A the sum of all data values, n the number of periods, and p the percentage increase relative to the first value, as defined in the report. yₙ = y₁ + (n − 1) · p y₁ A = ½ · [2y₁ + (n − 1) · p y₁] · n n = [−(2y₁ − p y₁) ± √((2y₁ − p y₁)² + 8p y₁A)] / (2p y₁), with n ≥ 0. Translation note: The findings and conclusions above are those reported by the student authors. The notation is retained from the original; the PDF provides the full derivation and context.