Question

1. A lifeguard has been given 500m of rope to enclose a rectangular swim area designated for toddlers and children. The swimming area will be accessed from the beach; therefore only three sides must be roped in. Determine the dimensions of the rectangle that would maximize the swimming area.

Answer 1

swimming area:

$S=xy$

perimeter:

$P=2(x+y)=500+x$

then:

$y=\frac{500+x}{2}-x$

$S=x(\frac{500+x}{2}-x)$

$S'(x)=\frac{500+x}{2}-x-x/2=250-x=0$

$x=250,y=125$

$S_{max}=250\cdot125=31250$ m²