Question

The sum of two numbers is 8. Find the smallest possible value for the sum of the cube of one number and the square of the other.

Answer 1

Please begin by representing these two numbers. You could choose the letters x and y to do that.

Using these letters, write an equation for "the sum of two numbers is 8."

Choose one of the numbers (doesn't matter whether you choose x or y). Square it.
Choose the other number. Cube it.

Add together the square and the cube.

Using your first equation, eliminate one of the two variables.

Assuming that you know calculus: take the first deriv. of this sum function and set the deriv. = to 0. Solve the resulting eq'n for whichever variable is left (x or y).

Know what to do next?

Check your results.

Answer 2

x^2+(-x)^3=8

Answer 3

So I got to this point

Answer 4

x = 1.716189

Answer 5

Idk how to do the derivative stuff, I don't know very much calculus. It's all very blurry

Answer 6

But what about that '8'? The sum of your two numbers is 8. You'll need to use this fact to eliminate one of the variables.

Sum is 8: x+y=8. Solve this for either x or y, please.

Answer 7

y= 0x + 8

Answer 8

-x

Answer 9

The 'sum of x and y' would be x+y.
Since the sum of x and y is 8, we can write x+y=8.

You cannot eliminate the '8', nor can you eliminate both variables.

Again I give you x+y=8. Solve this for x (altho' you could solve for y as an alternative).

Answer 10

x = -y + 8

Answer 11

Yes, that's right. You subtracted y from both sides.

What is the square of x, if x=-y+8? Here you are "squaring a binomial."

Answer 12

x^2=?

Answer 13

Square of the sum of two numbers a and b is (a+b)^2=a^2+2ab+b^2.
Square of the difference of two numbers a and b is (a-b)^2 = a^2 - 2ab -b^2.

Answer 14

Thus, the square of (8-y) is what?

Answer 15

(8-y)^2 = 8^2 - 28y - y^2?

Answer 16

Actually, the middle term is -16y. How did you get -28y?

Answer 17

2(8y)

Answer 18

I was just typing it out, I hadn't done any of the math yet

Answer 19

2*8=16, right?

I see. Can't write "28" for "2 times 8." Must write 2(8) or 2*8.

Answer 20

yes

Answer 21

"The sum of two numbers is 8. Find the smallest possible value for the sum of the cube of one number and the square of the other."

x+y=8, so x=-y+8. Correct. Thus, x^2=(-y+8)^2 = y^2 -16y +64. This is the "square of one number," and that number happens to be x. In terms of y, that number happens to be y^2 -16y +64.

OK with that? "square of one number"

Answer 22

We are dealing exclusively with 'y,' having solved x+y=8 for x. OK with that?

Square of one number (in terms of y) is y^2-16y+64.
Cube of one number (y) is y^3.

Write out the sum of "the square of one number and the cube of the other."

Answer 23

(y^2 - 16y + 65) + (y^3)

Answer 24

That's 64, not 65. Otherwise your expression is fine. That's "the sum of the square of one number and the cube of the other."

At this point we have to decide what to do next. If you're able to find the derivative of this function of yours, do it.

If you're not, then you could graph this function on a graphing calculator. You would see right away where the minimum value is and approximate the y value there.

Your choice?

Answer 25

sorry, typo

I'll graph it

Answer 26

or how could I find the derivative?

Answer 27

Before I explain that I need to ask just how long you have studied calculus. Do you remember the "power rule?"

Answer 28

Is it when you take the exponent, multiply it by the coefficient and subtract 1 from the exponent?

Answer 29

Yes.

What is the derivative with respect to y of y^3?

of y^2?

Answer 30

eeek, okay I understand it but I might get it wrong

3y^2?

Answer 31

2y?

Answer 32

yes. yes.

It's very important that you review the power rule in differentiation.

if f(y)=y^3 + y^2 - 16y + 64,
then the derivative will be
f '(y) = 3y^2 - 2y - 16.

Set that = to 0 now.

Are you able to solve this quadratic equation for y?

Answer 33

y = 8/3, -2 ?

Answer 34

Beg your pardon. That derivative is f '(y) = 3y^2 + 2y - 16.

One root is y=+2.

That agrees perfectly with my graph of f(y)= y^3 + y^2 - 16y + 64.
This graph shows a minimum at y=2.

Answer 35

Any questions about this process?

x+y=8,
x=-y+8
x^2 = y^2 -16y + 64

y^2 + y^3 = y^2 - 16y + 64 + y^2

Answer 36

Finding the derivative of that last function results in 3y^2 +2y -16. We set this = to 0 and solve for y. y=2. OK?

Answer 37

How would you find "the smallest possible value of the sum of the square of one number and the cube of the other?"

Answer 38

Hint: You already have a function f(y) for this. It is y^3 + y^2 - 16y + 64.

Answer 39

16?

by plugging in y?

Answer 40

Yes, plug in y=2: let y=2 in y^3 + y^2 - 16y + 64.

Answer 41

44?

Answer 42

Perfect. You know your stuff, really.

That's your answer.

Answer 43

y=2. You must also find x. Remember how x and y are related?

Answer 44

x = 6

Answer 45

perfect. I've just given you a medal.

Answer 46

Thank you for the help! You're the best!

Answer 47

Thank you so much. I love to love you. ;) Bye!

Answer 48

Haha bye :)