Question

find a single expression that represents the area of the outer ring of the circle if the area of the whole circle is represented by the expression \25x^{2}-12x - 9.\
a 18x^{2}-12x - 6
b 18x^{2}-12x - 12
c 32x^{2}-12x - 6
d 32x^{2}-12x - 12

Explanation:

Step1: Recall the relationship

The area of the outer - ring is the area of the whole circle minus the area of the inner circle.

Step2: Set up the subtraction

Let \(A_{outer - ring}\) be the area of the outer - ring, \(A_{whole}\) be the area of the whole circle, and \(A_{inner}\) be the area of the inner circle. So \(A_{outer - ring}=A_{whole}-A_{inner}\). Given \(A_{whole}=25x^{2}-12x - 9\) and \(A_{inner}=7x^{2}-3\).

Step3: Perform the subtraction

\((25x^{2}-12x - 9)-(7x^{2}-3)=25x^{2}-12x - 9 - 7x^{2}+3\).
Combine like - terms: \((25x^{2}-7x^{2})-12x+(-9 + 3)=18x^{2}-12x - 6\).

Answer:

A. \(18x^{2}-12x - 6\)