To determine the correct inequality that represents the given situation, we'll translate the problem statement into a mathematical expression step-by-step:
1. Define the variables:
- Let [tex]\( x \)[/tex] be the width of the photo.
2. Determine the dimensions of the cake:
- 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 twice its width. Since the width of the cake is [tex]\( x + 4 \)[/tex], the length of the cake is [tex]\( 2(x + 4) \)[/tex].
3. Express the area of the cake:
- The area of a rectangle is given by its length multiplied by its width. Therefore, the area of the cake can be calculated as:
[tex]\[
\text{Area} = \text{Length} \times \text{Width} = 2(x + 4)(x + 4)
\][/tex]
4. Set up the inequality:
- We are told that the area of the cake Wanda is currently working on is at least 254 square inches. Therefore, we set up the inequality as follows:
[tex]\[
2(x + 4)(x + 4) \geq 254
\][/tex]
5. Simplify the inequality:
- First, expand the expression inside the inequality:
[tex]\[
2(x + 4)^2 \geq 254
\][/tex]
- Next, expand [tex]\( (x + 4)^2 \)[/tex]:
[tex]\[
(x + 4)^2 = x^2 + 8x + 16
\][/tex]
- So, the inequality becomes:
[tex]\[
2(x^2 + 8x + 16) \geq 254
\][/tex]
- Distribute the 2:
[tex]\[
2x^2 + 16x + 32 \geq 254
\][/tex]
Therefore, the inequality that represents this situation is:
[tex]\[ 2x^2 + 16x + 32 \geq 254 \][/tex]
Matching with the given options, the correct answer is:
[tex]\[ \boxed{2x^2 + 16x + 32 \geq 254} \][/tex]
Hence, the correct answer is option B.
