Wanda is a cake designer with a specialty in rectangular silk screen photo cakes. For every cake that she makes, the width of the cake is 4 inches more than the width of the photo in the center of the cake, and the length of every cake is two times its width. The area of the cake Wanda is currently working on is at least 254 square inches.
If [tex]$x$[/tex] represents the width of the photo, which inequality represents this situation?
A. [tex]$x^2 + 8x + 16 \geq 254$[/tex] B. [tex][tex]$8x^2 + 64x + 128 \geq 254$[/tex][/tex] C. [tex]$x^2 + 4x \geq 254$[/tex] D. [tex]$2x^2 + 16x + 32 \geq 254$[/tex]
To determine the correct inequality, let's break down the problem mathematically step-by-step.
1. Define the variables: - Let [tex]\(x\)[/tex] represent the width of the photo in the center of the cake (in inches).
2. Express the dimensions of the cake: - The width of the cake is [tex]\(x + 4\)[/tex] inches (since the width is 4 inches more than the width of the photo). - The length of the cake is double the width of the cake. Hence, the length is [tex]\(2 \times (x + 4) = 2x + 8\)[/tex] inches.
3. Calculate the area of the cake: - The area of a rectangle is given by the product of its width and length. - So, the area [tex]\(A\)[/tex] of the cake is [tex]\((x + 4) \cdot (2x + 8)\)[/tex].
4. Set up the inequality: - Given that the area of the cake is at least 254 square inches, this can be translated into an inequality: [tex]\[
(x + 4)(2x + 8) \geq 254
\][/tex]
We can see that this matches the answer in the question. Therefore, the correct inequality that represents this situation is:
