In Exercises 53-62, some applications of inequalities are...

In Exercises $53-62,$ some applications of inequalities are shown.
The length $L$ and width $w$ (in yd) of a rectangular soccer field should satisfy the inequalities $110 \leq L \leq 120$ and $70 \leq w \leq 80$ Express the possible diagonal lengths $d$ as an inequality.

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Key Concepts

Inequalities

Inequalities express limits on the possible values of variables and are used to describe ranges or constraints in a problem. They allow us to define bounds within which quantities must lie, making them essential for formulating and solving problems that involve restricted values.

Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry that relates the lengths of the sides of a right triangle. In the context of rectangles, it is used to calculate the diagonal by taking the square root of the sum of the squares of the length and width.

Monotonicity of Functions

Understanding the behavior of functions, particularly the fact that the square function and the square-root function are monotonic (consistently increasing) for non-negative inputs, is crucial. This property ensures that when the input values are at their minimum or maximum, the outputs (such as the computed diagonal length) will correspond to the overall minimum or maximum possible values.

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