x^2+y=120 find y. what is the max area? (the problem is this: a rectangle and a fence to cover 3 sides, find area of the longest side.) (120 is the amount of fencing needed for all 3 sides)

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- anonymous

is the equation you mentioned is equation of area ?

- anonymous

yes

- anonymous

rearrange the equation,
y=120-x^2
to find critical point, you have to take the first derivative equals to 0.
dy/dx = -2x =0 thus x=0.
so now we have a critical point that we don't know it's a max or min point.
since you're having a quadratic equation with negative coefficient in front of x^2,
generally we know the curve and the critical point you got is a maximum point.
to find max area, pluck in the value of x=0 into your equation and you'll get y=120 which is the maximum area.

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- anonymous

so y=120, which is also the max, area?

- anonymous

i forgot to ask, is y the variable that represent the area this problem?

- anonymous

*area of this problem

- anonymous

well the problem is laid out like this:
a mall wants to add a restaurant to its outside in a rectangular shape. so 3 sides (one long side is against building so you dont use it. and they want to put fencing around the 3 sides which requires 120 feet of fencing.
you have to find:
1)length of all sides
which mean find the y (which i am using as the one long side)
and you have to find:
2)the maximum area.

- anonymous

does that make sense?

- anonymous

ah-ha.
so y is the length of all sides.
i think it should be 2x+y=120.
area of the restaurant =xy

- anonymous

Oh yeah I think so! but can you compute that?

- anonymous

rearrange the length equation,
y=120-2x,
substitute the y above into equation of area,
A= x(120-2x) = 120x -2x^2
now differentiate A with respect to x,
dA/dx = 120 -4x =0,
now you have a critical point, x=30.
again we have a quadratic point with negative coefficient on x^2.
x is maximum.
to find maximum area, pluck in x=30 into the equation of area and you'll get 1800.

- anonymous

so the max area is 1800 feet? and y=60?

- anonymous

yep :)

- anonymous

thanks so much!!

- anonymous

do you know this:
how many integer values of (a) can x^2+ax+6 be factored? what are they?

- anonymous

the highest order of x is two.
it's a quadratic function.
is the value of a given in the question ?

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