Question

Find two numbers who's difference is 10 and who's product is a minimum? can you walk me through the steps as well?

Answer 1

This problem involves both algebra and calculus.

Choose a letter to represent each of the two numbers. For example, choose x and y to be the two numbers.

Given that the 2 numbers are x and y, how would you write a formula for the "difference" of x and y? And what is that "difference" equal to?

Answer 2

\[x-y=10\]

Answer 3

?

Answer 4

that makes \(x=y+10\) so the product is \(y\times (y+10)\) or you could use \(x(x+10)\) as the product
you certainly don't need calculus to find the minimum of the quadratic
\[x^2+10x\]

Answer 5

matter of fact you don't need algebra at all

Answer 6

Yes. x-y = 10. Also, the area is A = x * y.

You'll need to solve x-y=10 for x and then substitute the result into A = x*y.
Would you please do that? satellite73 has already gotten there.

Think of how you could find the x-value at which you have the max. area

Answer 7

minimum is what you are looking for right?

Answer 8

you don't really need algebra at all
since you cannot tell the numbers apart, i.e. it is symmetric in \(x\) and \(y\) the minimum occurs when they are equidistant from 0, i.e. one is 5 and the other is -5
but you can use algebra to find the vertex of \(A=x^2+10x\) if you like

Answer 9

\[A=(x^2+10x)*x\]

Answer 10

?

Answer 11

I agree with satellite73's \[A=x^2+10x\]

Answer 12

so just\[x^2+10x\]

Answer 13

Erob: graph this: A-x^2 + 10x. You could then find the x-value at which you have a minimum just by looking at the graph.

Answer 14

Use the Draw utility, please.

Answer 15

can just one person explain because im getting confused on which step and where im suppose to put things? and start over?

Answer 16

All right. Hold a minute. I'm tieing up another matter.

Answer 17

k

Answer 18

Find two numbers who's difference is 10 and who's product is a minimum? can you walk me through the steps as well?

choosing x and y to represent the 2 numbers, x-y=10 is the "difference." Thus, x=y+10. Do you believe that?

Answer 19

yes

Answer 20

And you want to MINIMIZE the product, P, of the two numbers, correct?

Answer 21

yeah

Answer 22

Good. Then your product, P = x*y, becomes (y+10)*y after we substitute x = y+10.

Thus, P(y) = (y+10)*y. OK with this?

Answer 23

ok got that so y^2+10y

Answer 24

Yes. Label it. P(y) = y^2 + 10y.

Just by looking at this, we can see that the graph is that of a parabola that opens up. Correct?

Answer 25

yeah

Answer 26

Therefore, the graph has a minimum value. We have to find that minimum value. Know how to do that?

Answer 27

no....

Answer 28

We could either graph the function P(y)=y^2+10y and look to see which y value results in a minimum P, or we could use the formula y=-b/(2a) to find the y-coordinate of the vertex. The vertex represents the minimum in this case. What approach would you prefer?

Answer 29

I already have the graph so we can do it that way.

Answer 30

Nice work. Please estimate what y value results in P being at its MINIMUM.

Answer 31

-24?

Answer 32

Erob: This seems to be a graph of a function different from ours. Ours is P(y)=1y^2+10y. Are you sure that's what you've graphed?

Answer 33

Yeah hold on let me check

Answer 34

yeah it is

Answer 35

we can just use the equation it's no problem

Answer 36

I'm much more comfortable with your new graph. Thank you. Notice that this graph represents the PRODUCT of x and y. If you'll go down to the vertex (the minimum), you'll see that the y-value of the vertex is -5 (halfway in between -10 and 0). Agreed?

Answer 37

ok

Answer 38

So it appears that our solution is y=-5, x= y + 10 = 5.

Answer 39

ok so x=5 and y=-5

Answer 40

Look what happens: If x= 5 and y=-5, their product is -25.
if x=6 and y=-4, their product is -24
so it really does appear that we get the smallest product when x=5 and y=-5. OK with this?

Answer 41

yeah thank for the help

Answer 42

Happy that we've reached a satisfactory solution! Hope to work with you again. Take care, Erob! bye.