4. A student is given a budget of $\$ 10000$ to produce a rectangular banner for a school function. The length and width (in metres) of the banner must be integers. If each metre in length costs $\$ 330$ while each metre in width costs $\$ 450$, what is the maximum area (in $\mathrm{m}^{2}$ ) of the banner that can be produced?
(1 mark)
一名學生要為學校的活動製作一幅長方形的宣傳橫額, 開支上限為 $\$ 10000$ 。橫額的長度和闊度（以米為單位）必須是整數。若製作橫額每長一米收費 $\$ 330$, 每闊一米收費 $\$ 450$, 問可製作的橫額的面積最大是多少平方米？
4. 165
4. Let the length and width of the banner be $x \mathrm{~m}$ and $y \mathrm{~m}$ respectively. According to the question, $x, y$ are positive integers with $330 x+450 y \leq 10000$. Thus the area of the banner is
$$
x y=\frac{(330 x)(450 y)}{(330)(450)} \leq \frac{1}{(330)(450)}\left[\frac{330 x+450 y}{2}\right]^{2} \leq \frac{1}{(330)(450)}\left[\frac{10000}{2}\right]^{2}<169 \mathrm{~m}^{2}
$$
by the AM-GM inequality. Note that we must have $x \leq 30$ and $y \leq 22$.
If the area is $168 \mathrm{~m}^{2}$, then we have the possibilities $(x, y)=(12,14) ;(14,12)$ and $(24,7)$, yet none of them satisfies $330 x+450 y \leq 10000$. Since 167 is prime and the only factors of 166 are $1,2,83$ and 166 , we easily see that the area cannot be $167 \mathrm{~m}^{2}$ nor $166 \mathrm{~m}^{2}$ either.

Finally, we see that $x=15$ and $y=11$ satisfies all conditions and give an area of $165 \mathrm{~m}^{2}$. Thus the maximum area of the banner is $165 \mathrm{~m}^{2}$.