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The manager of a 139139-unit apartment complex finds that all units are rented at a price of $2025$2025 per month. Research shows that for each $45$45 increase in rent, one additional unit remains vacant. How much should he charge for rent in order to bring in maximum revenue and how many units are rented then?

Question

The manager of a 139139-unit apartment complex finds that all units are rented at a price of 202520252025 per month. Research shows that for each 454545 increase in rent, one additional unit remains vacant. How much should he charge for rent in order to bring in maximum revenue and how many units are rented then?

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Solution

To solve this problem, we need to first understand the relationship between the rent price, the number of rented units, and the total revenue.

Step 1: Define the variables Let's denote the number of 45increasesinrentasx.Therefore,therentpricewillbe45 increases in rent as x. Therefore, the rent price will be 2025 + 45x and the number of rented units will be 139 - x.

Step 2: Formulate the revenue function The total revenue (R) is the product

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