Question

问一道ap微积分的题目，求解答。

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?
直接给答案也可以。

Answer 1

楼主你好！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：英文解答。

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.