# Max Profit

#### mathkid3

A real Estate office handles a 50-unit apt. complex. When the rent is \$580/mo. all units are occupied. For each \$40 increase in rent, however, an avg of one unit becomes vacant. Each occupied unit requires an avg of \$45 per month for service and repairs. What rent should be charged to obtain a maximum profit? I need help setting up the problem TY! Last edited by a moderator: #### MarkFL ##### Administrator Staff member The dollar sign character is used for rendering LaTeX here, that is why your post looks that way. We need to find a function which describes the number of units occupied as a function of the rent charged. Let the rent be$\displaystyle 580+40x$We also know that for each increase of 1 in x, we get a decrease of 1 in U, the number of units occupied. Hence:$\displaystyle \frac{dU}{dx}=-1$where$\displaystyle U(0)=50$Can you now solve this IVP to find$\displaystyle U(x)$? You may choose to simply use the point-slope formula to find this linear function. #### mathkid3 ##### New member I may need a little further assistance in what to differentiate first one I have seen like this #### MarkFL ##### Administrator Staff member First, we want to find the number of units occupied as a function of the number of 40 dollar increases in rent. So, either integrate the ODE I gave, or more simply use the point-slope formula to find the function. You know a point (0,50) and the slope m = -1. #### mathkid3 ##### New member im getting y=-x+50 now what coach ? #### MarkFL ##### Administrator Staff member Okay, so we now have:$\displaystyle U(x)=50-x$Now, we know profit is revenue minus cost. Can you state the profit as a function of U and x?$\displaystyle P(U,x)=?$Note: don't worry, we will get rid of U to get the profit as a function of one variable in the next step. #### mathkid3 ##### New member Im sorry Mark I dont follow #### MarkFL ##### Administrator Staff member For the revenue: The money coming in is the number of units occupied times the rent charged per unit. What is this product? For the cost: The total cost is the number of units occupied times the cost of upkeep per unit. What is this product? #### mathkid3 ##### New member P = 580(50)-50(45) ? #### MarkFL ##### Administrator Staff member That's only true if x = 0. Recall we defined the rent as:$\displaystyle r(x)=580+40x$And instead of using 50 for the units occupied, we want to use$\displaystyle U(x)$. What do you get now? #### mathkid3 ##### New member hey is the profile pic u? P = (580+40z)(50)(45) ? #### MarkFL ##### Administrator Staff member No, that is one of my intellectual heroes, Dr. Ed Witten, the only physicist to win the Fields Medal in mathematics, and arguably the world's foremost expert in M-theory. Now, revenue is number of units occupied ($\displaystyle U(x)=50-x$) times rent charged ($\displaystyle r(x)=580+40x$), so we have the revenue:$\displaystyle R(x)=(50-x)(580+40x)$And cost is units occupied ($\displaystyle U(x)=50-x$) times average cost per unit (45 dollars), so we have the cost function:$\displaystyle C(x)=45(50-x)$So, what is the profit function$\displaystyle P(x)$? #### mathkid3 ##### New member P = (50-z)(580+40z)-45(50-z) ? #### MarkFL ##### Administrator Staff member Yep, now I would factor, expand, then optimize. You don't have to expand, you could use the product rule to differentiate. Your choice. edit: What kind of number do we require x to be? Last edited: #### mathkid3 ##### New member P ' = -40z^2 + 1465z + 26750 ? am I done? #### MarkFL ##### Administrator Staff member That's not the derivative, that's the expanded profit function. Now differentiate and equate to zero...but be careful as z must be what kind of number? Hint: look at the function defining the number of units occupied. #### mathkid3 ##### New member wait is Z = -1 ? #### MarkFL ##### Administrator Staff member No, but how did you determine that value? #### mathkid3 ##### New member I was just thinking by the way u said it that z = -1 is it that z must be positive ? #### MarkFL ##### Administrator Staff member First, think about the fact that U must be a non-negative integer in [0,50], so what kind of number must z be and what is the feasible domain? #### mathkid3 ##### New member Mark, sorry I fell asleep on couch last night while chatting with you on the forum. I had to get coffee in me to start to re what we were doing last night ok I have questions. #1 how did I get so lost in the problem from the start?? I think it was from asking your help you introduced so variables in the equation that didn't match with the textbook. Prob just overwhelmed me from the start. I left u with giving you the expanded profit function instead of taking the derV and setting to zero let me try again... P ' (x) = -80x + 14652 solving the org P function for x I get x = -1387.75 and x = -1452.25 This is for defining the feasable domain (Is that correct Mark?) anyway... I wish we could start from scratch in this problem and u could walk me through it again. I do not like it when I get lost in a problem. I need to understand why we set it up the way we did and why I am struggling with a concept(s) Thanks bud! #### MarkFL ##### Administrator Staff member Okay, we are given: A real Estate office handles a 50-unit apt. complex. When the rent is \$580/mo. all units are occupied.

For each \$40 increase in rent, however, an avg of one unit becomes vacant. Each occupied unit requires an avg of \$45 per month for service and repairs.

What rent should be charged to obtain a maximum profit?

Here's the way I would look at it:

Let z represent the number of \$40 increases in rent, that is, the monthly rent per unit is:$\displaystyle r(z)=580+40z$We are told that for each increase of 1 in z, there is a decrease of 1 in the number of units occupied, which we can state as:$\displaystyle u(z)=50-z$The monthly revenue is the total amount of rent collected, which is equal to the number of units occupied times the monthly rent per unit:$\displaystyle R(z)=u(z)\cdot r(z)=(50-z)(580+40z)=20(50-z)(29+2z)$The total monthly cost is the average cost per unit times the number of units occupied:$\displaystyle C(z)=45u(z)=45(50-z)$Now, the monthly profit is the monthly revenue minus the monthly cost:$\displaystyle P(z)=R(z)-C(z)=20(50-z)(29+2z)-45(50-z)=5(50-z)(4(29+2z)-9)=5(50-z)(107+8z)$Now, I am going to let you use differentiation, while I am going to use the fact that we have a parabolic profit function opening downward whose axis of symmetry will be midway between the roots, and thus the vertex (maximum point) will be on this axis. The roots are:$\displaystyle 107+8z=0\,\therefore\,z=-\frac{107}{8}\displaystyle 50-z=0\,\therefore\,z=50$The axis of symmetry is then the line (using the mid-point formula):$\displaystyle z=\frac{-\frac{107}{8}+50}{2}=\frac{293}{16}\$

See if you can get this critical value using the calculus. After this, we'll deal with the fact that we should treat z as a discrete variable rather than a continuous one.