The manager of a large apartment complex knows from experience that 80 units will be occupied if the rent is 472 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 4 dollar increase in rent. Similarly, one additional unit will be occupied for each 4 dollar decrease in rent. What rent should the manager charge to maximize revenue?
d unit/drent= 1unit/4dollars rent.
revenue=80+x)(472-4x)
drevenue/dx=-4(80+x) + (472-4x)=0
240+4x-472=-4x
8x=232
x=29 units, so rent should be 472-4*29
check that.
To determine the rent that will maximize revenue, we need to understand the relationship between rent and the number of occupied units. From the given information, we know that at a rent of $472 per month, there will be 80 occupied units.
From the market survey, we know that for each $4 increase in rent, one additional unit will remain vacant. So if we increase the rent by $4, there will be 79 occupied units. Similarly, for each $4 decrease in rent, one additional unit will be occupied. If we decrease the rent by $4, there will be 81 occupied units.
To find the optimal rent, we need to consider the revenue generated at each rent level. Revenue is calculated by multiplying the number of occupied units by the rent.
At a rent of $472 per month, the revenue is 80 * $472 = $37,760.
If we increase the rent by $4, the new rent would be $476, and the revenue would be 79 * $476 = $37,604.
If we decrease the rent by $4, the new rent would be $468, and the revenue would be 81 * $468 = $37,908.
Comparing the three scenarios, we can see that decreasing the rent by $4 generates the highest revenue of $37,908.
Therefore, the manager should charge a rent of $468 per month to maximize revenue.