Question

Find two positive numbers whose sum is 18 and the product of the first number and the
square of the other is a maximum.

Answer 1

in this i am getting 2 x values did i do somthing wrong or.. ?

Answer 2

What did you get? It's possible if you are getting two different answers they are both correct, or even possible that they are the same answer.

Answer 3

the x values i get 6 and 18

Answer 4

am i right ??

Answer 5

6 + 18 do not sum up to 18.

Answer 6

true

Answer 7

Here is our system of equations:
\[\Large\sf 18=x+y\]\[\Large\sf P=xy^2\]We want to rewrite P in terms of only x or y before we take a derivative.\[\Large\sf P=(18-y)y^2\]\[\Large\sf P=18y^2-y^3\]Understand what I did there?
Now we can differentiate P with respect to y, to look for max/min points.

Answer 8

i almost did the same thing but i did 18-x is that okey ??

Answer 9

Did you set it up as 18 = x + x?

Answer 10

\[\Large\sf P=xy^2\]\[\Large\sf P=x(18-x)^2\]You can do it this way if you want,
but you have to expand out the square or apply the product rule which is a little bit annoying.

Answer 11

i get 6 and 18 after i take the derv and simpligying is that what u get too ?

Answer 12

\[\Large\sf P=x(18-x)^2\]\[\Large\sf P'=(18-x)^2-2x(18-x)\]Setting our first derivative equal to zero to find max/min,\[\Large\sf 0=(18-x)^2-2x(18-x)\]Solving for x gives,\[\Large\sf x=18,\qquad x=6\]These are the two possible values for `ONE OF OUR NUMBERS`.
They are NOT both of our numbers.
This is only the x value, yes?

And only one of these values maximizes the product, the other is a minimum probably.

Answer 13

beacuse the question says maximum so i choose 18 ??

Answer 14

So if we let x=18,

18 = x + y
18 = 18 + y

What does y have to be in this case?

Answer 15

Subtract 18 from each side... what do you get for y..?

Answer 16

Hullo..? +_+

Answer 17

ohh sorry

Answer 18

y is 0

Answer 19

Ok so that is one possibility.
x=18, y=0.

Does that maximize the product of the two numbers?

P=xy^2
P=18*0^2

Answer 20

i dont get what u mean by mzximize the product

Answer 21

The product of 18 and 0 is 0, right?

This is kinda tricky to explain,
Lemme see if I can make some sense out of it for you.

If we could pick any two positive numbers to multiply together...
They should give us a positive value, right?
So if they multiplied together and gave us zero...

That value is the smallest of the possible outcomes we would expect right?
Not the maximum.

So we found the x value that `minimizes` the product x*y^2.

Does that make... a little sense, maybe? D:

Answer 22

So we chose the wrong x value.
x=18 is a minimum of our P function.
We want the other value.

Answer 23

Refer to the Mathematica calculation attached.