Parabola in a Parallelogram

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A parabola is given with the x-coordinate of the minimum point, and two points on the parallelogram shown. The points are joined and then a parallel line is drawn that is a tangent to the parabola. The task is to determine the ratio of the area bounded by the upper line and the parabola, to the remaining area.

There are some suggestions made for how to proceed and the mathematics involved encompasses both differentiation, determining the equation of a line given two points, finding the equation of a tangent, and integration.

Having worked through the problem, students are then shown a ‘common sense’ method of determining the solution by considering a shear of the parabola. In addition, an alternative formal method is investigated to solve the problem without using integration.

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