Question

in the accompanying diagram, the width of the inner rectangle is represented by ( x - 3 ), and the length by ( x + 3 ). the width of the outer rectangle is represented by ( 3x - 4 ), and the length by ( 3x + 4 ).
a. write an expression (in expanded form) to represent the area of the larger rectangle.
b. write an expression (in expanded form) to represent the area of the smaller rectangle.

Explanation:

Step1: Recall the area formula for a rectangle

The area of a rectangle is \(A = length\times width\).

Step2: Find the area of the larger rectangle

For the larger rectangle, length \(l = 3x + 4\) and width \(w=3x - 4\).
Using the formula \(A=(3x + 4)(3x - 4)\).
By the difference - of - squares formula \((a + b)(a - b)=a^{2}-b^{2}\), where \(a = 3x\) and \(b = 4\).
So \(A=(3x)^{2}-4^{2}=9x^{2}-16\).

Step3: Find the area of the smaller rectangle

For the smaller rectangle, length \(l=x + 3\) and width \(w=x - 3\).
Using the formula \(A=(x + 3)(x - 3)\).
By the difference - of - squares formula \((a + b)(a - b)=a^{2}-b^{2}\), where \(a=x\) and \(b = 3\).
So \(A=x^{2}-3^{2}=x^{2}-9\).

Answer:

A. \(9x^{2}-16\)
B. \(x^{2}-9\)