Equation

Saumyojit

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IF a/b = b/c=c/d then solve (b^3 + c^3 + d^3)/(a^3+b^3+c^3)

I dont want an approach using k.

I have started like this :
a/b = b/c =>
ac=b^2

b/c=c/d => bd =c^2

How will i minimize (b^3 + c^3 + d^3)/(a^3+b^3+c^3) ???
 

Mr. Bland

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Dec 27, 2019
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What exactly is it that needs to be solved? a, b, c and d can be solved using only the proportions, and the other expression isn't an equation.
 

Saumyojit

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What exactly is it that needs to be solved? a, b, c and d can be solved using only the proportions, and the other expression isn't an equation.
It needs to be minimised to a/d .
How can I bring it without using k .
 

firemath

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I don't see a k.......
 

Saumyojit

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I don't see a k.......
most of the people would solve it using a/b = b/c=c/d = k
a=bk.....
I dont want to use k
i want to approach it like this
a/b = b/c =>
ac=b^2

b/c=c/d => bd =c^2
 

firemath

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I see......I was taught that formula differently.
So you want a compressed answer in variable terms?
 

Saumyojit

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I see......I was taught that formula differently.
So you want a compressed answer in variable terms?
see i know that
ac=b^2

b/c=c/d => bd =c^2

how will i substitute
ac=b^2
bd =c^2 in (b^3 + c^3 + d^3)/(a^3+b^3+c^3) ?
 

firemath

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a=b^2/c
c=b^2/a
b=c^2/d
d=c^2/b

Do you follow?
 

Dr.Peterson

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Messages
6,203
IF a/b = b/c=c/d then solve (b^3 + c^3 + d^3)/(a^3+b^3+c^3)

I dont want an approach using k.

I have started like this :
a/b = b/c =>
ac=b^2

b/c=c/d => bd =c^2

How will i minimize (b^3 + c^3 + d^3)/(a^3+b^3+c^3) ???
Why not use the k method, which seems simplest?

I think you mean, not "minimize" (which means, find the lowest value of), but "simplify".

see i know that
ac=b^2

b/c=c/d => bd =c^2

how will i substitute
ac=b^2
bd =c^2 in (b^3 + c^3 + d^3)/(a^3+b^3+c^3) ?
One way is to express c and d in terms of a and b: c = b^2/a, d = c^2/b = b^3/a^2. Make those substitutions, and simplify.

Using a/b = b/c = c/d = k (or, as I prefer, b/a = c/b = d/c = k) is a lot less complicated.
 

Saumyojit

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Messages
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Why not use the k method, which seems simplest?

I think you mean, not "minimize" (which means, find the lowest value of), but "simplify".


One way is to express c and d in terms of a and b: c = b^2/a, d = c^2/b = b^3/a^2. Make those substitutions, and simplify.

Using a/b = b/c = c/d = k (or, as I prefer, b/a = c/b = d/c = k) is a lot less complicated.
I have substituted c = b^2/a, d = c^2/b = b^3/a^2 .
but i am stuck after 2 steps and i cannot further minimize it .
Please show me the whole steps using this method : c = b^2/a, d = c^2/b = b^3/a^2
 

Dr.Peterson

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Please show your work, so I can see where you are stuck! I want to help you learn to get yourself unstuck.
 

Saumyojit

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Messages
113
(b^3 + c^3 + d^3)/(a^3+b^3+c^3) =(abc +bdc+d^3)/(a^3+abc+bdc)=then what?

b^3=abc

c^3=bdc
 

Dr.Peterson

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That's not the substitution I suggested. Why did you say, "I have substituted c = b^2/a, d = c^2/b = b^3/a^2"? You didn't.

The idea is to express everything in terms of a and b only, so that you can simplify the expression. For example, c^3 will become (b^2/a)^3 = b^6/a^3, and d^3 will become (b^3/a^2)^3 = b^9/a^6. You'll have a complex fraction that you can simplify by multiplying numerator and denominator by the LCD.
 

Saumyojit

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USING UR METHOD :
(b^3 + c^3 + d^3)/(a^3+b^3+c^3) = {b^3+b^6/a^3 +b^9/a^6 }/{a^3+b^3+b^6/a^3)}
=> {b^3 *a^6 +b^3*a^3+b^9}/a^6 * a^3 / {a^6+a^3 * b^3 + b^6} =>( b^3 *a^6 +b^3*a^3+b^9)/(a^6+a^3 * b^3+ b^6) * 1/a^3

after doing lcm

=>b^3 (a^6 +a^3+b^6)/a^3(a^6+a^3 * b^3+ b^6) then what??
 

Dr.Peterson

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Then, since the parenthesized factors in numerator and denominator look so similar, you should check your work to see if maybe they are really identical and can be cancelled!
 

Saumyojit

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Messages
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I have tried every possible way but not minimised
 

Dr.Peterson

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Messages
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Please do as I said, and check your work on factoring the numerator. Is it really true that

{b^3+b^6/a^3 +b^9/a^6 } = {b^3 *a^6 +b^3*a^3+b^9}/a^6 ?​

A major part of doing mathematics well is to learn to check your work and catch errors.
 

Subhotosh Khan

Super Moderator
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USING UR METHOD :
(b^3 + c^3 + d^3)/(a^3+b^3+c^3) = {b^3+b^6/a^3 +b^9/a^6 }/{a^3+b^3+b^6/a^3)}
=> {b^3 *a^6 +b^3*a^3+b^9}/a^6 * a^3 / {a^6+a^3 * b^3 + b^6} =>( b^3 *a^6 +b^3*a^3+b^9)/(a^6+a^3 * b^3+ b^6) * 1/a^3

after doing lcm

=>b^3 (a^6 +a^3+b^6)/a^3(a^6+a^3 * b^3+ b^6) then what??
Did you REVIEW your work and are you certain that above expression is correct?
 

Dr.Peterson

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@Saumyojit: To encourage you to continue, I will add that there is just one number wrong, and after fixing it you will be almost finished.
 

Saumyojit

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Jan 21, 2020
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