问一道ap微积分的题目,

问一道ap微积分的题目,
The manager of a large apartment complex knows from experience that 90 units will be occupied if the rent is 372 dollars per month.A market survey suggests that,on the average,one additional unit will remain vacant for each 3 dollar increase in rent.Similarly,one additional unit will be occupied for each 3 dollar decrease in rent.What rent should the manager charge to maximize revenue?
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这道题的程度与pre calculus或者intermediate algebra的方程应用题相当,大致意思如下：经理发现当租金是每月372美元时,90单位的住房可以租出去.在此基础上每月的租金每上升或下降3美元就会导致租出去住房的数量减少或增加1个单位.问经理应该把月租金定为多少以获取最高总利益?
设总利益为P,月租金为x美元.
P= x*[90-1/3*(x-372)]=x*(214-1/3*x)=214*x-1/3*x^2=-1/3*(x^2-642*x)=-1/3*[(x-321)^2-103041]
-1/3*(x-321)^2+34347 (x>0);
根据二次函数的增减性可知,当月租金x=321（美元）的时候,总利润最高,为34347美元.
附：
Suppose that the total monthly revenue is P US dollars,and that the rent is x US dollars per month.
P= x*[90-1/3*(x-372)]=x*(214-1/3*x)=214*x-1/3*x^2=-1/3*(x^2-642*x)=-1/3*[(x-321)^2-103041]
-1/3*(x-321)^2+34347 (x>0);
As can be seen from the monotonity of function P=-1/3*(x-321)^2+34347 (x>0),when x=321,the maximum of P is reached.
Thus,the manager should charge $321 per month to receive the maximal revenue.