Find two positive numbers satisfying the given requirements

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jkh1919

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I worked on this problem for an hour or so earlier, and came up short every time. I am hoping someone can show me the right way to do this. I'll show the problem and the steps I've taken to solve it.

The product is 192 and the sum of the first plus three times the second is a minimum.

xy = 192
x+3y = 0 (a tutor told me to set this equal to zero, but I'm not sure why, I just did it because he told me to)

y = 192 / x

Quotient Rule:

u = 192
v = x
uprime = 0
vprime = 1

x + (192 / x) = y

1 + x(0) - 192(1) / x^2 = 1/1 - 192 / x^2

yprime = x^2 - 192 / x^2

This is as far as I've gotten, to explain quickly, I set y equal to 192/x and found the derivative using the quotient rule. I don't know what to do next, my guess is find the critical numbers (which I know one would be + or - the sq rt of 192, but don't know much more after that).
 
jkh1919 said:
I worked on this problem for an hour or so earlier, and came up short every time. I am hoping someone can show me the right way to do this. I'll show the problem and the steps I've taken to solve it.

The product is 192 and the sum of the first plus three times the second is a minimum.

xy = 192
x+3y = 0 .................. not necessarily true.. (a tutor told me to set this equal to zero, but I'm not sure why, I just did it because he told me to)

y = 192 / x

Quotient Rule:

u = 192
v = x
uprime = 0
vprime = 1

x + (192 / x) = y .........................................How's that???!!!

1 + x(0) - 192(1) / x^2 = 1/1 - 192 / x^2

yprime = x^2 - 192 / x^2

This is as far as I've gotten, to explain quickly, I set y equal to 192/x and found the derivative using the quotient rule. I don't know what to do next, my guess is find the critical numbers (which I know one would be + or - the sq rt of 192, but don't know much more after that).
Click to expand...

let's assume

A = x + 3*y

and you need to minimize A.

Then

A = x + 576/x

dA/dx = 1 - 576/x2

Now continue......

.
 
jkh1919 said:
x + 3y = 0

tutor told me to set [x+3y] equal to zero
Click to expand...

Was that your dad? ;)

You already know that both x and y are greater than zero (i.e., both positive). You should have realized that positive numbers cannot possibly sum to zero, and then rejected that advice.

I didn't try to follow your application of the quotient rule, but here is a shortcut: not all ratios require the quotient rule.

EG:

y = 125/x

is the same as

y = 125 x^(-1)

Use the Power Rule

y=(-1)(125)x-1-1y = -125 x^(-2)

is the same as

y` = -125/x^2

:cool:
 
Ok, I'm still not sure how to solve this problem. What it is asking for anyway. The 2 choices are 1st number and 2nd number. My school and the book I have give an explanation only someone already proficient in calculus could possibly understand, and if there's anything I'm not, it's proficient in calculus.
 
jkh1919 said:
dA/dx = 1 - 576^(-2)

This is where I'm not following. Why the minus sign?
Click to expand...

My example answers your question, I think, but you need to have already learned the Power Rule to understand my example.

Do you know that 1/x is the same as x^(-1) ?

In the power x^(-1), the exponent is negative.

Do you understand the Power Rule?

When you bring the exponent -1 "down in front" to multiply the term 576 x^(-1 - 1), the result is -576 x^(-2)

Did you try to work the derivative?

I do not quite follow where you're not following. :cool:
 
jkh1919 said:
I worked on this problem for an hour or so earlier, and came up short every time. I am hoping someone can show me the right way to do this. I'll show the problem and the steps I've taken to solve it.

The product is 192 and the sum of the first plus three times the second is a minimum.

xy = 192
x+3y = 0 (a tutor told me to set this equal to zero, but I'm not sure why, I just did it because he told me to)
Click to expand...

To be fair to him, I suspect what your tutor told you was to take the derivative first and set that equal to 0.

This is what comes of doing something "because he told me to" with knowing why. You misunderstand or, with a slightly different problem do the same thing as before without recognizing the difference makes that wrong. If a tutor or teacher tells you to do something and you are not sure why, ask why!

y = 192 / x

Quotient Rule:

u = 192
v = x
uprime = 0
vprime = 1

x + (192 / x) = y

1 + x(0) - 192(1) / x^2 = 1/1 - 192 / x^2

yprime = x^2 - 192 / x^2

This is as far as I've gotten, to explain quickly, I set y equal to 192/x and found the derivative using the quotient rule. I don't know what to do next, my guess is find the critical numbers (which I know one would be + or - the sq rt of 192, but don't know much more after that).
Click to expand...
 
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