Question 105520
If we represent length and width as L and W, respectively:
The rectangular field must have an area of at least 500sqm (to make it easier, we will use equal sign instead of the greater-than-or-equal sign:
eq1{{{LW = 500}}}
Perimeter is 100m:
eq2{{{2L + 2W = 100}}}
First, isolate W in eq2:
{{{2W = 100 - 2L}}}
{{{W = 50 - L}}}
Then, substitute (50 - L) for W in eq1:
{{{L(50-L) = 500}}}
{{{50L - L^2 = 500}}}
{{{L^2 - 50L + 500 = 0}}}
{{{L^2 - 50L = -500}}}
{{{L^2 - 50L + 625 = 125}}}
{{{(L - 25)^2 = 125}}}
{{{L - 25 = +-5sqrt(5)}}}
{{{L = 25 +-5sqrt(5)}}}
Since L > W:
{{{L = 25 + 5sqrt(5)}}}
Since lessening the value of L until 25 makes the area greater:
{{{25m <= L <= (25 + 5sqrt(5))m}}}