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Advanced Math / Nonlinear functions Difficulty: Medium

The function $f left parenthesis w right parenthesis equals 6 w squared$ gives the area of a rectangle, $in square feet ({ft}^{2})$ , if its width is $w ft$ and its length is $6$ times its width. Which of the following is the best interpretation of $f left parenthesis 14 right parenthesis equals 1,176$ ?

If the width of the rectangle is $14 ft$ , then the area of the rectangle is $1,176 ft squared$ .

If the width of the rectangle is $14 ft$ , then the length of the rectangle is $1,176 ft$ .

If the width of the rectangle is $1,176 ft$ , then the length of the rectangle is $14 ft$ .

If the width of the rectangle is $1,176 ft$ , then the area of the rectangle is $14 ft squared$ .

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Explanation

Choice A is correct. The function $f$ gives the area of the rectangle, in $ft Superscript 2$ , if its width is $w ft$ . Since the value of $f left parenthesis 14 right parenthesis$ is the value of $f (w)$ if $w equals 14$ , it follows that $f left parenthesis 14 right parenthesis equals 1,176$ means that $f (w)$ is $1,176$ if $w equals 14$ . In the given context, this means that if the width of the rectangle is $14 ft$ , then the area of the rectangle is $1,176 ft squared$ .

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from interpreting $f (w)$ as the width, in $ft$ , of the rectangle if its area is $w {ft}^{2}$ , rather than as the area, in $ft Superscript 2$ , of the rectangle if its width is $w ft$ .