Mathematics

Problem

A rectangular plot of land has an area of

1800\,\text{m}^2

. The length of the plot exceeds twice its width by

10\,\text{m}

. (i) Take the width to be

x\,\text{m}

and express the length in terms of

x

. Hence form a quadratic equation in

x

that represents the given information. (ii) Calculate the discriminant of this quadratic equation and comment on the nature of its roots. (iii) Solve the quadratic equation obtained in part (i) and determine the dimensions of the plot correct to two decimal places. (iv) Using your answer from part (iii), find the perimeter of the plot, again correct to two decimal places.

Intuitive explanation

This problem connects a real-world constraint—a fixed area and a relationship between dimensions—to a quadratic equation. The key insight is translating the phrase "length exceeds twice the width by 10 m" into an algebraic expression, then using the area formula to create an equation in one variable. Once you have the quadratic, the discriminant tells you whether real solutions exist, and the quadratic formula gives you the actual dimensions.

Intuitive explanation

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