Optimizing Rent and Revenue: Real Estate and Wholesale Pricing Strategies

[nyr]

Okay so I need help with 2 problems. One I did a bunch of work for but I am not sure if my answers right, and the second I'm not sure how to start it.
The first problem:

A real estate office handles 50 apartment units. When the rent is $720 per month, all units are occupied. However, on average, for each $40 increase in rent, one unit becomes vacant. Each occupied unit requires an average of $48 per month for services and repairs. what rent should be charged to obtain the maximum profit.


My work:

I set up an equation
R= (50-x)(720+40x)-48(50-x)
R'=1328-80x
critical number at x=16.6

50-16.6=33.4
R(16.6)/33.4=1336
$1336 should be charged to obtain the maximum profit.

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The second problem:
When a wholesaler sold a certain product at $25 per unit, sales were 800 units per week. After a price increase of $5, the average number of units sold dropped to 775 per week. Assume that the demand function is linear and find the price that will maximize the total revenue.

I have no idea how to set up this problem

 

[lippyka]

. Any help would be appreciated.The equation you need to set up is a linear equation of the form y=mx+b, where y is the number of units sold per week, m is the slope or rate of change, and b is the y-intercept. To find the price that will maximize total revenue, you need to find the point where the slope of the line becomes zero. To do this, you can use the formula m=-(Δy/Δx), where Δy is the change in y and Δx is the change in x. Plugging in the given information, you get m=-(800-775)/(25-5)=-25/20=-1.25. Now you can solve for b using one of the given points (e.g. (25, 800)). You get b=825. Your equation becomes y=-1.25x+825. To find the point where the slope becomes zero, you need to solve the equation 0=-1.25x+825, which yields x=660. This means that the price that will maximize total revenue is $660.