To determine the correct inequality that represents the given situation:
1. Let [tex]\( x \)[/tex] denote the width of the photo in the center of the cake.
2. According to the problem:
- The width of the cake is 4 inches more than the width of the photo.
Therefore, the width of the cake is [tex]\( x + 4 \)[/tex].
- The length of the cake is two times its width.
Therefore, the length of the cake is [tex]\( 2(x + 4) \)[/tex].
3. The area of the cake is calculated by multiplying its length and width:
[tex]\[
\text{Area of the cake} = \text{Length} \times \text{Width}
\][/tex]
Substituting the expressions for length and width:
[tex]\[
\text{Area of the cake} = 2(x + 4) \times (x + 4)
\][/tex]
4. The area of the cake must be at least 254 square inches, so we set up the inequality:
[tex]\[
2(x + 4)(x + 4) \geq 254
\][/tex]
5. Simplify the expression inside the inequality:
[tex]\[
2(x + 4)^2 \geq 254
\][/tex]
6. Expand the quadratic expression:
[tex]\[
2(x^2 + 8x + 16) \geq 254
\][/tex]
7. Distribute the 2:
[tex]\[
2x^2 + 16x + 32 \geq 254
\][/tex]
Therefore, the correct inequality is:
B. [tex]\(2x^2 + 16x + 32 \geq 254\)[/tex]
So the correct answer is option B.
