twelfth grade calculus, optimization problems

Forums: High School, Calculus, Area, Derivatives, Profit
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I actually have multiple questions in regards to the optimization unit in calculus, but I’ll start with this one.

A rectangular piece of land is to be fenced using two kinds of fencing. Two opposite sides will be fenced using standard fencing that costs $6/m, while the other two sides will require heavy-duty fencing that costs $9/m. What are the dimensions of the rectangular lot of greatest area that can be fenced for a cost of $9000?

A real estate office manages 50 apartments in a downtown building. When the rent is $900 per month, all the units are occupied. For every $25 increase in rent, one unit becomes vacant. On average, all units require $75 in maintenance and repairs each month. How much rent should the real estate office charge to maximize profits?

I am so lost, any help would be appreciated

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@gardnstate,

Start with simple equations.

x = cheap side
y = expensive side
6x + 9y = 9000
Area = xy

Next combine the two equations.
y = 1000 - 2x/3
Area = x(1000 -2x/3)

Finally, take the derivative of that area equation to find the maximum.

For the second one
income = rent x occupied apartments - repairs
occupied apartments = 50 - (rent - 900)/25
repairs = 50 x $75

Combine and find where income is a maximum

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@gardnstate,

Same approach different process

a) 6x+9y=C
A=xy
C=9000

6dx+9dy=0 dx/dy=-9/6
xdy+ydx=0 dx/dy=-x/y
so x=3y/2

6(3y/2)+9y=C
18y=C
y=C/18=9000/18=1000/2=500 ft
x=3y/2=3(500)/2=750 ft

b) C=75*50
I=R*O-C
O=50-(R-900)/25

I=R[50-(R-900)/25]-C
I=50R-R^2/25+36R-C
I=-R^2/25+86R-C
I=1/25[-R^2+2150R-25C]
dI/dR=-2R+2150=0
R=2150/2=$1075. 7 units empty

Rap

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@engineer,

I'm so sorry, I know the question I'm about to do is not about this topic, but I could not make it on that forum and I chose one at random. As stated by my name is Italian, but this year 'I would come to finish high school in America.
but there are many things that I do not know: for example, how much is the school? how we teach? I am here in Italy I have to start the year leceo clear, if I move to America will also start from the third year?
aspect reply Smile

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twelfth grade calculus, optimization problems

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