Select the correct answer.

Wanda is a cake designer specializing in rectangular silk screen photo cakes. For every cake she makes, the width of the cake is 4 inches more than the width of the photo in the center, and the length of each 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]2 x^2+16 x+32 \geq 254[/tex]
B. [tex]8 x^2+64 x+128 \geq 254[/tex]
C. [tex]x^2+4 x \geq 254[/tex]
D. [tex]x^2+8 x+16 \geq 254[/tex]



Answer :

Let's translate the situation described into mathematical terms.

1. Width of the cake: The width of the cake is 4 inches more than the width of the photo. If [tex]\( x \)[/tex] represents the width of the photo, then the width of the cake is [tex]\( x + 4 \)[/tex].

2. Length of the cake: 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. Area of the cake: The area of a rectangle is given by the product of its width and length. Therefore, the area of the cake is:
[tex]\[ \text{Area} = \text{Width} \times \text{Length} = (x + 4) \times 2(x + 4) \][/tex]

4. Simplify the expression for the area:
[tex]\[ \text{Area} = (x + 4) \times 2(x + 4) = 2(x + 4)^2 \][/tex]

5. Inequality representing the area constraint: The area of the cake must be at least 254 square inches, so we write the inequality as:
[tex]\[ 2(x + 4)^2 \geq 254 \][/tex]

The correct inequality that represents the situation is:
[tex]\[ 2(x + 4)^2 \geq 254 \][/tex]

This is most similar to option D if we consider the expanded form incorrectly. However, since we are looking for the direct form that matches [tex]\(2*(x + 4)^2 \geq 254\)[/tex]:

None of the listed options directly match [tex]\(2(x + 4)^2 \geq 254\)[/tex]. But the closest correct transformation of this would be in option D considering the sources.

Therefore, the correct answer is:
E. None of the listed options are direct fits.