Question

Beth has 2000 feet of fencing available to enclose a rectangular field. one side of the field lies along a river, so only three sides require fencing. express the area a of the rectangle as a function of x, where x is the length of the side parallel to the river.

Answer 1

Answer:   $f(x)=\dfrac{2000x-x^2}{2}$

Step-by-step explanation:

Perimeter (P) is the total length of the fencing which covers 2 sides of one length and 1 side of another length (because the side along the river does not need fencing).  So, P = 2L+ x  where L is the length and x is the width.

P = 2000 and P = 2L + x   →   2000 = 2L + x

$\text{Solve for L:}\\\\2000 = 2L + x\\\\2000-x=2L\\\\\dfrac{2000-x}{2}=L$

Area (A) is length x width  →  A = L × x

$\text{Use substitution to replace L with}\ \dfrac{2000-x}{2}\\\\A = f(x)=\bigg(\dfrac{2000-x}{2}\bigg)x\\\\\\.\qquad \qquad =\dfrac{2000x-x^2}{2}$