Question

A local company manufactures netbook computers. Their profit function is given by this equation: y=-x^2+120x+3000. where x is the number of netbooks produced in one day and y is profit (in dollars) the company makes that day. What is the maximum profit that the company can make?
(Find the vertex of the parabola)

Answer 1

do you take calc?

Answer 2

no

Answer 3

dy/dx = -2x + 120

-2x + 120 = 0

x = 60

Answer 4

x= 60, y = 6600 profit max

Answer 5

So is that the answer?

Answer 6

yeah, the vertex is at (60,6600) and the parabola faces down

Answer 7

ok thanks!! And can i ask you one more question?

Answer 8

Or no.

Answer 9

im out.. cyas

Answer 10

Sorry, your equation is \(y=-x^2+120x+3000\) and your vertex would be at \(x-\dfrac{-b}{2a}\)

Answer 11

\[x=\frac{-b}{2a}\]

Answer 12

If you have not learned calculus, then you simply need to find the x coordinate of your vertex, use that to find your y-coordinate of the vertex, and you will have the coordinate points for your vertex.

Answer 13

so it would be X=-(-1) over 2(-1)?

Answer 14

Let's check.\[x=-\frac{b}{2a} = \frac{-120}{-2(1)} =~?\]

Answer 15

ohh forgot the -120 bit

Answer 16

Analyze your formula :\[y=\color{red}{-x^2}\color{blue}{+120x}\color{green}{+3000}\]\[y=\color{red}{a}x^2\color{blue}{+b}x\color{green}{+c}\]

Answer 17

ok so we a which is -1 and b=120, c=3000

Answer 18

Here im showing you that your equation matches the general formula for a quadratic equation.

Answer 19

yes! :)

Answer 20

completing the square would be more easy here

Answer 21

yes!! ok

Answer 22

Let's try a different method this time @ganeshie8 hahaah

Answer 23

you had breakup wid completing the square is it hmm

Answer 24

Yeah I agree it would be easier, but first a person needs to learn which values match up to the values in a quadratic :P

Answer 25

Yeah I diid lol

Answer 26

So what would be your x-value, @Lovelucy19 ?

Answer 27

why do you need to learn that for completing the square ?

Answer 28

60?

Answer 29

You don't, it's just a different method for finding the vertex I think.I'll show her compelting the square afterwards :P

Answer 30

OK - vertex form gives geometric intuition of the profit function, this -b/2a thingy is more like memorizing and algebra

Answer 31

Good, \(x=60\).

Now we can write our equation, \(y=-x^2+120x+3000\) as \(f(x)=-x^2+120x+3000\)

therefore,since we found our \(x\) value, we can plug that in for our function, \(f(60) = -(60)^2 +120(6)+3000\)

Answer 32

Question why don't we put anything on the C part?

Answer 33

Is there an x in the c part of our function?

Answer 34

if there is not, then we cannot put anything there :P

Answer 35

ohh ok i was getting a little confused, but im good now

Answer 36

Oh, good. :) So what do we get for f(60)?

Answer 37

isn't that just 60 still or would that be -60?

Answer 38

For the first term? \(-(60)^2 = -(60 \cdot 60) =~?\)

Answer 39

3,600

Answer 40

And then you multiply the negative to it.

Answer 41

-3600

Answer 42

remember, if the negative is outside, \(-(60)^2\) vs \((-60)^2\), the first one will give you \(-3600\) and the second one will give you 3600. that is because you multiply a -1 to a function that you first square, the first one, and for the second one, you are doing this: \((-60) \cdot (-60)\)

Answer 43

If that makes any sense, so you have -3600 for the first term .
then add it to \(120 \cdot 6\). What does that give you?

Answer 44

720

Answer 45

What I meant is: -3600 + (120 * 6) =?

Answer 46

and yes, 120 * 6 = 720 \(\checkmark\)

Answer 47

ok so itwouuld be -2880

Answer 48

Yes :)

Answer 49

-2880 +3000?

Answer 50

120

Answer 51

Yes ) so y=120

Answer 52

Therefore our vertex is \(\boxed{(60,120)}\)

Answer 53

Thanks so much for your help with this I honestly didn't know what to do, but now with your help i feel like i can pass the test next week!!!! thanks again for everything!!!!

Answer 54

yaaaay!!