Question

Find two positive real numbers whose product is a maximum. (Enter your answers as a comma-separated list

Answer 1

the sum of the first and twice the second is 20

Answer 2

Hey Lisa! :)
What type of class is this for?
Can we use Calculus Optimization methods?

Answer 3

Hey! and no this is precalc with algebra

Answer 4

Ooo that's no fun >.<
I'm trying remember how to do these optimize probems without calculus...
Hmmm

Answer 5

There is one problem I did where it said the sum is equal to 130
So I set up an equation of x+y=130 and then I divided both sides by 2 and got 65,65 as my answer

Answer 6

I doubt my reasoning is right but I got the correct answer

Answer 7

first isolate y in `x+y=130` to get `y = -x+130`

Answer 8

so `x*y` turns into `x*(-x+130) = -x^2 + 130x`

Answer 9

finding the vertex of `f(x) = -x^2 + 130x` will help you find the max product and the two values to get the max product

Answer 10

Oh wow okay that makes sense how 65 is the answer

Answer 11

`-x^2 + 130x` is in the form `ax^2 + bx + c `

a = -1
b = 130

compute `h = -b/(2a)` to get `h = 65`. Then plug this into `-x^2 + 130x` to find the product

Answer 12

Okay I see. How would I set up the equation for the first problem?

Answer 13

This one? `the sum of the first and twice the second is 20`

Answer 14

Yes

Answer 15

would it be x+2y=20?

Answer 16

correct

Answer 17

then you can isolate x to get x = -2y+20

so x*y = (-2y+20)*y = -2y^2 + 20y

then treat y as x (they're both variables, so why not)

find the vertex of -2x^2 + 20x

Answer 18

= 5

Answer 19

so y is really 5 since I replaced y with x

y = 5

x = -2y+20
x = -2*5+20
x = -10+20
x = 10

Answer 20

ohhh wow I understand. Thank you so much for helping me @jim_thompson5910 !!

Answer 21

you're welcome

Answer 22

Do you mind helping me with another question?

Answer 23

sure

Answer 24

I have to write the quadratic function x^2-10x in standard form

Answer 25

do you mean in vertex form?

Answer 26

no the directions say standard.....I think you have to convert it into vertex to get the standard

Answer 27

does your book say that standard form is ax^2 + bx + c? or y - k = a(x-h)^2 ?

Answer 28

first one

Answer 29

well x^2-10x is already in standard form

look at how it matches up to ax^2 + bx + c

in this case
a = 1
b = -10
c = 0

Answer 30

no...when I plug it in it tells me its wrong
so I guess the second one

Answer 31

ok so you need to find the vertex

Answer 32

you first need to compute h = -b/(2a)

Answer 33

h=5

Answer 34

yes, now plug that back into the given function to find y

Answer 35

25-50= -25
y=-25

Answer 36

yep the vertex is (5,-25)

Answer 37

so (h,k) = (5,-25)

h = 5
k = -25

Answer 38

a = 1
h = h
k = -25

plug that into `y - k = a(x-h)^2`

Answer 39

-50=(x-5)^2...?

Answer 40

you should get `y+25 = (x-5)^2`

Answer 41

okay then what?

Answer 42

that's the answer in `y-k = a(x-h)^2` form