The length of the pond is 39.75 m and the width is 29.75 m.
Explanation:
The width would be given by (40-x), where x is the width of the grass strip. The length would be (50-x). We know that their area is 1184; the area is found by multiplying the width and length:
(40-x)(50-x)=1184.
Multiplying these binomials, we have
40*50-40*x-50*x-x*(-x)=1184
2000-40x-50x+x²=1184
Combining like terms, we have
2000-90x+x²=1184.
We subtract 1184 from both sides:
2000-90x+x²-1184 = 1184-1184
816-90x+x²=0.
Use the quadratic formula to solve this:
[tex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}
\\
\\x=\frac{--90\pm \sqrt{(-90)^2-4(1)(816)}}{2(1)}
\\
\\x=\frac{90\pm \sqrt{8100-3264}}{2}=\frac{90\pm \sqrt{4836}}{2}
\\
\\x=\frac{90\pm 69.5}{2}=\frac{90+69.5}{2}\text{ or }\frac{90-69.5}{2}
\\
\\x=\frac{159.5}{2}\text{ or }\frac{20.5}{2}=79.75\text{ or }10.25[/tex]
Since the entire length of the rectangle is only 50 m, the width of the strip around the pond cannot be 79.75; this means the width of the grass strip must be 10.25. This makes the length of the pond 50-10.25=39.75 and the width of the pond 40-10.25=29.75.
