5. Sadinie is asked to create a rectangular enclosure using $40 \mathrm{~m}$ of fencing so that the side-lengths, in metres, of the enclosure are all prime numbers. What is the maximum possible area of the enclosure?
A $51 \mathrm{~m}^{2}$
B $84 \mathrm{~m}^{2}$
C $91 \mathrm{~m}^{2}$
D $96 \mathrm{~m}^{2}$
E $99 \mathrm{~m}^{2}$
Solution
C

Let the length and width, in metres, of the rectangle be $l$ and $w$, respectively. We need consider only cases where $l<w$. Since the side length of the fence is $40 \mathrm{~m}, 40=2 \times(l+w)$. So $l+w=20$.

Both $l$ and $w$ are prime numbers. There are 2 possible pairs $(l, w)=(3,17),(7,13)$.
Maximum possible area is $(7 \times 13) \mathrm{m}^{2}$ which is $91 \mathrm{~m}^{2}$