问一道ap微积分的题目,

工科留学生为你解答!
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.