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Advanced Math / Nonlinear functions Difficulty: Hard

Square P has a side length of $x$ inches. Square Q has a perimeter that is $176$ inches greater than the perimeter of square P. The function $f$ gives the area of square Q, in square inches. Which of the following defines $f$ ?

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Explanation

Choice A is correct. Let $x$ represent the side length, in inches, of square P. It follows that the perimeter of square P is $4 x$ inches. It's given that square Q has a perimeter that is $176$ inches greater than the perimeter of square P. Thus, the perimeter of square Q is $176$ inches greater than $4 x$ inches, or $4 x plus 176$ inches. Since the perimeter of a square is $4$ times the side length of the square, each side length of Q is $\frac{4 x + 176}{4}$ , or $x plus 44$ inches. Since the area of a square is calculated by multiplying the length of two sides, the area of square Q is $(x + 44) (x + 44)$ , or $left parenthesis x plus 44 right parenthesis squared$ square inches. It follows that function $f$ is defined by $f (x) = {(x + 44)}^{2}$ .

Choice B is incorrect. This function represents a square with side lengths $left parenthesis x plus 176 right parenthesis$ inches.

Choice C is incorrect. This function represents a square with side lengths $left parenthesis 176 x plus 44 right parenthesis$ inches.

Choice D is incorrect. This function represents a square with side lengths $left parenthesis 176 x plus 176 right parenthesis$ inches.