My professor won't be back for a week and we have an exam tomorrow. The sub was going over a problem towards this question but I barely understood her. Tried to speak to her after class but she was very impatient. I need help please.
The manager of a large apartment complex knows from experience that 120 units will be occupied if the rent is 294 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 7 dollar increase in rent. Similarly, one additional unit will be occupied for each 7 dollar decrease in rent. What rent should the manager charge to maximize revenue?
revenue = rent * apts
if there are x $7 increases, then we have
rent = 294+7x
units rented = 120-x
So, the revenue will be
(294+7x)(120-x) = -7x^2 + 546x + 35280
Recall that the vertex of a parabola is at x = -b/2a, which in this case is
x = 546/14 = 39
So, with 39 rent increases,
rent = 294 + 7*39 = $567
and there will be 120-39 = 81 units rented.
OK so we have two equations from the word problem and we need to equal them our and basically find the 0 ?
apparently you stopped reading before finishing things. Read on past where it says
So, the revenue will be ...
No I read the work I over looked it by accident. i see you multiplied it into a polynomial function.
And you have used the vertex in order to find x and that's the only way to find x? or is there an alternative ?
To find the rent that the manager should charge to maximize revenue, we need to understand the relationship between the number of occupied units and the rent amount. From the given information, we know that at a rent of $294 per month, 120 units will be occupied. Additionally, for every $7 increase in rent, one unit will remain vacant, and for every $7 decrease in rent, one unit will be occupied.
To solve this problem, we need to approach it systematically. Here's a step-by-step process on how to find the optimal rent:
1. Define the variables: Let's represent the rent as 'R' and the number of occupied units as 'U.'
2. Determine the relationship between rent and the number of occupied units: From the given information, we can establish that for every $7 increase in rent, one unit will remain vacant, and for every $7 decrease in rent, one unit will be occupied. Thus, we can write the relationship as:
U = 120 - (R - 294)/7
3. Calculate the revenue: The revenue can be calculated by multiplying the rent by the number of occupied units. Hence:
Revenue = R * U
4. Substitute the value of 'U' from step 2 into the revenue equation in step 3:
Revenue = R * (120 - (R - 294)/7)
5. Simplify the revenue equation: Multiply 'R' by the values inside the brackets and simplify to get the revenue equation in terms of 'R' only.
6. Find the maximum revenue: To find the rent that maximizes revenue, we need to find the value of 'R' that maximizes the revenue equation. This can be done by finding the derivative of the revenue equation with respect to 'R' and setting it equal to zero. Then solve the resulting equation for 'R'.
7. Once you find the value of 'R' that maximizes revenue, substitute it back into the revenue equation to find the maximum revenue.
Please note that without specific numbers for the rent increase or decrease, it is not possible to provide a numerical solution for the problem. However, following the steps above will help you find the optimal rent using the given information.