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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 $x$ represents the width of the photo, which inequality represents this situation?


A$2 x^2+16 x+32 \geq 254$
B$x^2+8 x+16 \geq 254$
C$8 x^2+64 x+128 \geq 254$
D$x^2+4 x \geq 254$

Verified Answer :

  • Define the width of the cake as \(W = x + 4\) and the length as \(L = 2(x + 4)\).
  • Calculate the area \(A = L \times W = 2(x + 4)^2\).
  • Expand the area expression to get \(2(x^2 + 8x + 16) = 2x^2 + 16x + 32\).
  • Conclude that the inequality representing the situation is \(\boxed{2x^2 + 16x + 32 \geq 254}\).

Explanation

  1. Defining Variables and Relationships

To represent the situation described for the cake, we first identify the given variables and their relationships. Let \(x\) be the width of the photo (in inches). The width of the cake, \(W\), is given as 4 inches more than the width of the photo, which can be written as \(W = x + 4\). The length of the cake, \(L\), is twice its width, so \(L = 2W = 2(x + 4)\).

  1. Formulating the Area Equation

The area of a rectangular cake is calculated by multiplying its length by its width, so the area \(A\) is given by \(A = L \times W\). Substituting the expressions we found for \(L\) and \(W\), we get:
\[A = [2(x + 4)] \times (x + 4) = 2(x + 4)^2.\]

  1. Constructing the Inequality

Now, we expand the expression \(2(x + 4)^2\):
\[2(x + 4)^2 = 2(x^2 + 8x + 16) = 2x^2 + 16x + 32.\]
Since the area of the cake must be at least 254 square inches, we set up the inequality \(A \geq 254\), which results in:
\[2x^2 + 16x + 32 \geq 254.\]

  1. Final Conclusion

By comparing our derived inequality \(2x^2 + 16x + 32 \geq 254\) with the given choices, we see that it matches the first option. Thus, the inequality that represents the situation is \(2x^2 + 16x + 32 \geq 254\). The final answer is \(\boxed{2x^2 + 16x + 32 \geq 254}\).

Examples

In real-life scenarios like construction, manufacturing, or design, we often need to express area or volume constraints using algebraic inequalities. For example, if a designer needs to fit a logo of a specific size onto a product with custom dimensions, setting up an area constraint helps ensure the final product meets size requirements. By expressing all dimensions as a function of one variable, we can build a mathematical model to solve for the necessary bounds or ranges that ensure the project meets its specifications.

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