Question

A wire of length \(x\) is bent into the shape of a square. (a) Express the perimeter \(p\) of the square as a function of \(x\). (b) Express the area \(A\) of the square as a function of \(x\).

Short answer

(a) \(p(x) = x\). (b) \(A(x) = \frac{x^2}{16}\).

Step-by-step solution

Understand the Problem

A wire of length \(x\) is bent into the shape of a square. We need to express the perimeter and area of the square as functions of \(x\).

Express Perimeter as Function of \(x\)

The perimeter of a square is the sum of all its sides. If the length of the wire is \(x\) and it is bent into a square, then this entire length forms the perimeter. Hence, we have:\[ p(x) = x \]

Determine Side Length of the Square

Since a square has four equal sides and the total perimeter is \(x\), each side length \(s\) can be found by dividing the perimeter by 4:\[ s = \frac{x}{4} \]

Express Area as Function of \(x\)

The area of a square is given by the square of its side length. Using the side length from Step 3:\[ A(x) = s^2 = \left(\frac{x}{4}\right)^2 = \frac{x^2}{16} \]

Key concepts

Perimeter Function

Understanding the perimeter function is key to solving problems involving shapes and lengths. The perimeter of a shape is the total length around it. In this exercise, a wire of length \( x \) is bent into a square. The primary task is to express the perimeter of the square as a function of this wire length.

Area Function

The area function is another important component when dealing with shapes. The area of a square is the amount of space it occupies. To find the area function for our square, we rely on the side length we derived earlier. Given that each side of the square is \( \frac{x}{4} \), we can calculate the area as \( A(x) = s^2 \). Inserting our side length, we get: \ A(x) = \left( \frac{x}{4} \right)^2 = \frac{x^2}{16} \.

Side Length Calculation

To proceed with finding the perimeter and area, calculating the side length is crucial. We know that a square has four equal sides. If the total perimeter of our square is \( x \), each side length \ (s) \ is \ \frac{x}{4} \. This is derived by simply dividing the perimeter by the number of sides. Hence, \ s = x/4 \. This simple calculation forms the basis for expressing the area as a function of the given wire length.