Answer
(a) $l= \frac{600}{w}$ , $l=75-2w$
(b) $w=11.6 ft, 25.9 ft$
(c) $l= 23.2$ and $l= 51.9$
Work Step by Step
Length(l) x width(w) =600 $\implies l= \frac{600}{w}$ ...(1)
(a) Total length of the playground$=2w+l$
Thus, $2w+l=75 ft$
so, $l=75-2w$
(b) From equation (1), we have
$(\frac{600}{w})=75-2w$
$w^2-\frac{75}{2}w=-300$
Compare it with the standard form of quadratic equation $ax^2+bx+c$, we have $a=1, b=-\frac{75}{2}$
To complete the square, add $\dfrac{(-\frac{75}{2})^2}{4}=\frac{5625}{16}$ on both sides.
$w^2-\frac{75}{2}w+\frac{5625}{16}=-300+\frac{5625}{16}$
$\implies (w-\frac{75}{4})^2=\frac{825}{16}$
Neglect negative sign, we have
$\implies (w-\frac{75}{4})=7.2$
$w=11.6 ft, 25.9 ft$
(c) $l= \frac{600}{w}$ and $l= \frac{600}{w}$
$l= \frac{600}{11.6}=23.2$ and $l= \frac{600}{25.9}=51.9$
