For all positive values of P, W, L, and A, consider the family of rect

Official Explanation

For all positive values of P, W, L, and A, consider the family of rectangles having perimeter P feet, width W feet, length L feet, and area A square feet.

In the first column of the table, select the expression in terms of P and W that is equivalent to A, and in the second column of the table select the expression in terms of P and W that is equivalent to L. Select only two expressions, one in each column.

Strategize

For any rectangle, A = LW, and P = 2L + 2W. Dividing both sides of the latter equation by 2 yields P/2 = L + W, and subtracting W from both sides yields \([\frac{P}{2} -W]\) = L. Therefore L = \([\frac{P}{2} -W]\) and
A = LW

= \((\frac{P}{2} -W)W\)

Alternatively, observe that, since A = LW, the expression for A is equivalent to the expression for L with an additional factor, W. And the only two expressions in the list that have this relationship are C, \((\frac{P}{2}-W)\), and E, \((\frac{P}{2} -W)W\)

RO1, A: Strategize

The correct answer is E, \((\frac{P}{2} -W)W\)

RO2, L: Strategize

The correct answer is C, \((\frac{P}{2} -W)\)