Question

Find the pair of numbers whose sum is 48 and whose product is a maximum.

Answer 1

\(\large\color{black}{ \displaystyle y+x=48 }\)

\(\large\color{black}{ \displaystyle f(x,y)=y\times x }\)

Answer 2

\(\large\color{black}{ \displaystyle y+x=48 }\) >>> \(\large\color{black}{ \displaystyle y=48-x }\)


\(\large\color{black}{ \displaystyle f(x,y)=y\times x }\)
\(\large\color{black}{ \displaystyle f(x)=(48-x)\times x }\)

Answer 3

So by finding the absolute maximum of this function, you will get one of the numbers, and then plug it into y+x=48 to find the other number.

Answer 4

or, by logic you can get the answer as well.
because you know that (for instance)

20 * 20 is the greatest product from the addition of 2 numbers that has to be equal to 40.

Answer 5

you want to do the logic way, or the math way?

Answer 6

math way please :)

Answer 7

sure.
\(\large\color{black}{ \displaystyle f(x)=(48-x)\times x }\)

can you expand the parenthesis for me?

Answer 8

48x - x^2?

Answer 9

yes.

Answer 10

\(\large\color{black}{ \displaystyle f(x)=- x^2+48x }\)

Answer 11

normally to find the maximum of a function you would take the derivative, but in a case of a parabola (like your's) you do not need to do this.

Answer 12

you just need to re-write the parabola in a vertex form.

Answer 13

I am lagging so hard... sorry

Answer 14

no worries haha

Answer 15

\(\large\color{black}{ \displaystyle f(x)=-x^2+48x }\)
\(\large\color{black}{ \displaystyle f(x)=-(x^2-48x) }\)


now focus on what is in the parenthesis!

What do you have to add inside the parenthesis to make a perfect square trinomial inside the parenthesis?

Answer 16

I'm not sure...

Answer 17

do you know what a perfect square trinomial ?

Answer 18

is*

Answer 19

like a^2 + 2ab + b^2? because my teacher was really vague about this... sorry

Answer 20

yes.

Answer 21

for example.
x^2+2x+1
because it is same as (x+1)^2

Answer 22

but the basic thing here is, that for any
\(\large\color{black}{ \displaystyle x^2-{\rm \color{red}{b}}x }\)
to make it a perfect square, you would need to add \(\large\color{black}{ \displaystyle \left(\frac{{\rm \color{red}{b}}}{2} \right)^2 }\)

\(\large\color{black}{ \displaystyle x^2-{\rm \color{red}{b}}x+\left(\frac{{\rm \color{red}{b}}}{2} \right)^2 }\) is a perfect square trinomial.

Answer 23

ooh okay

Answer 24

in your case you have:

\(\large\color{black}{ \displaystyle x^2-{\rm \color{red}{48}}x }\)

so what number do you have to add ?

Answer 25

you don't need to calculate it, just say what it is (like how to get it)

Answer 26

(48/2)^2?

Answer 27

yes, or 24^2

Answer 28

but we will go ahead and calculate 576

Answer 29

So you had,

\(\large\color{black}{ \displaystyle f(x)=-(x^2-{\rm \color{red}{48}}x) }\)

Answer 30

now, we will add this number, but we will also subtract it, not to change the value of the right side.

\(\large\color{black}{ \displaystyle f(x)=-(x^2-{\rm \color{red}{48}}x~~~~+\left(\frac{{\rm \color{red}{48}}}{2} \right)^2~-~\left(\frac{{\rm \color{red}{48}}}{2} \right)^2~~) }\)

Answer 31

we will take out the negative fraction out of parenthesis, so it will be positive on the outside.

\(\large\color{black}{ \displaystyle f(x)=-(x^2-{\rm \color{red}{48}}x~~~~+\left(\frac{{\rm \color{red}{48}}}{2} \right)^2~) ~+~\left(\frac{{\rm \color{red}{48}}}{2} \right)^2}\)

Answer 32

So, you will have:

\(\large\color{black}{ \displaystyle f(x)=-(x-{\rm \color{blue}{24}})^2 +576}\)

Answer 33

is this parabola opening up or down?

Answer 34

down?

Answer 35

yes

Answer 36

that means it will have a maximum, but not a minimum.
The maximum here, is the vertex.

Answer 37

anytime, you have
\(\large\color{black}{ \displaystyle f(x)=-(x-{\rm \color{blue}{h}})^2 +k}\)
that means the vertex is \(\large\color{black}{ \displaystyle ({\rm \color{blue}{h}}, k)}\)
where h is the x-value at which the maximum occurs, and k is the y-value i.e. what the maximum is (saying - how high it gets).

Answer 38

so, your maximum is occurring at ?

Answer 39

h?

Answer 40

yes, but if you look a bit up, what is h in this case?

Answer 41

oh 24?

Answer 42

look at our new vertex form function,
\(\large\color{black}{ \displaystyle f(x)=-(x-{\rm \color{blue}{24}})^2 +576}\)
that I posted a bit ago.

Answer 43

24?

Answer 44

yes.

Answer 45

So you found one of the 2 numbers.

Answer 46

(you found the x)

Answer 47

so y = 576?

Answer 48

you know (from the very beginning) that y+x=48

Answer 49

oh nvm

Answer 50

so if x= 24, then y is also 24?

Answer 51

yes

Answer 52

THANK YOU SO MUCH!!!!!!

Answer 53

oh my gosh I was struggling with this but this helped sososos much! :)

Answer 54

the logic part is shorter.

lets take number \(\large\color{black}{ \displaystyle 16}\)

you know that
\(\large\color{black}{ \displaystyle 8\times 8}\)
is greater than any of the following:
\(\large\color{black}{ \displaystyle 7\times 9}\)
\(\large\color{black}{ \displaystyle 6\times 10}\)
\(\large\color{black}{ \displaystyle 5\times 11}\)
\(\large\color{black}{ \displaystyle 4\times 12}\)
\(\large\color{black}{ \displaystyle 3\times 13}\)
\(\large\color{black}{ \displaystyle 2\times 14}\)
\(\large\color{black}{ \displaystyle 1\times 15}\)

Answer 55

I will show what everything is equal to
\(\large\color{black}{ \displaystyle 8\times 8 =64}\)
\(\large\color{black}{ \displaystyle 7\times 9 =63}\)
\(\large\color{black}{ \displaystyle 6\times 10 =60 }\)
\(\large\color{black}{ \displaystyle 5\times 11 = 55}\)
\(\large\color{black}{ \displaystyle 4\times 12 = 48}\)
\(\large\color{black}{ \displaystyle 3\times 13 = 39}\)
\(\large\color{black}{ \displaystyle 2\times 14 = 24}\)
\(\large\color{black}{ \displaystyle 1\times 15 = 15}\)

Answer 56

(I am looking at the maximum product of two numbers that have a sum of 16)

Answer 57

And you can see that as the numbers get away from each other more and more
(like 2 and 24) the get smaller and smaller (than 8 and 8).

Answer 58

I mean like (2 and 12)

Answer 59

So, if 2 numbers a and b have a sum of C. And you want to find the maximum product, then a and b would each be C/2.

Answer 60

anyways, I guess I am overwhelming.
(won't post the calculus prove)

Answer 61

No, this was amazing! Thank you soso much! :)

Answer 62

you welcome!