## Spring Garden

Each spring, a $12$ meter by $12$ meter garden has its length increased by $2$ meters but its with decreased by $50$ centimeters. What will be the maximum attainable area of the garden?
Source: NCTM Mathematics Teacher, October 2006

Solution
Area in square meters per each year starting from year $0$
$0\!:\!(12)(12)$
$1\!:\!(12+2)(12-.5)=(12+1(2)(12-1(.5)$
$2\!:\!(12+2+2)(12-.5-.5)=(12+2(2))(12-2(.5)$
$3\!:\!(12+2+2+2)(12-.5-.5-.5)=(12+3(2))(12-3(.5))$
$\cdots$
In general in year $x$, the area equals
$(12+x(2))(12-x(.5))=-x^2+18x+144$
The graph of this quadratic expresion is shown below

The area reaches a maximum of $225$ square meters in year $x=9$.

Answer: $225$ square meters