Find the third proportional if first, second and fourth proportional are (i) 6, 18 and 54 (ii) (x+y)^2, x^3 + y^3 and x + y (iii) a^2 - b^2, a + b and a - b (iv) a^3 + b^3, a^2 - a... Find the third proportional if first, second and fourth proportional are (i) 6, 18 and 54 (ii) (x+y)^2, x^3 + y^3 and x + y (iii) a^2 - b^2, a + b and a - b (iv) a^3 + b^3, a^2 - ab + b^2 and a + b.
Understand the Problem
The question is asking to find the third proportional for given sets of numbers and expressions. This is a mathematical concept dealing with proportions and requires identifying relationships between the numbers or expressions provided.
Answer
1. The third proportional for \( 6, 18, 54 \) is \( 54 \).
Answer for screen readers
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For numbers ( 6, 18, 54 ), the third proportional is ( 54 ).
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For ( (x+y)^2, x^3+y^3 ), and ( x+y ), the third proportional depends on specific values of ( x ) and ( y ).
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For ( a^2-b^2, a+b ), and ( a-b ), and related sums, the third proportional requires calculating expressions respective to ( a ) and ( b ).
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For ( a^3+b^3, a^2-ab+b^2 ), and ( a-b ), calculations follow similarly.
Steps to Solve
- Finding the third proportional for numbers 6, 18, and 54
To find the third proportional ( x ) of the numbers 6 and 18, we set up the proportion:
$$ \frac{6}{18} = \frac{18}{x} $$
Cross-multiplying gives:
$$ 6x = 18 \cdot 18 $$
- Calculating the value of ( x )
Now simplify the equation:
$$ 6x = 324 $$
Dividing both sides by 6:
$$ x = \frac{324}{6} $$
- Final calculation
Calculating the value:
$$ x = 54 $$
- Finding the third proportional for ( (x+y)^2, x^3+y^3 ) and ( x+y )
Set up the proportion:
$$ \frac{(x+y)^2}{x^3+y^3} = \frac{x^3+y^3}{x+y} $$
Cross-multiplying gives:
$$ (x+y)^2(x+y) = (x^3+y^3)(x^3+y^3) $$
- Expanding and simplifying
Expand both sides:
$$ (x+y)^3 = (x^3+y^3)^2 $$
- Using the identity on the right
Recognize that:
$$ x^3 + y^3 = (x+y)(x^2 - xy + y^2) $$
So, substituting gives:
$$ (x+y)(y+y^2)^2 $$
The calculations and simplifications depend on specific values of ( x ) and ( y ).
- Finding the third proportional for ( a^2-b^2, a+b ) and ( a-b )
Using the proportion:
$$ \frac{a^2-b^2}{a+b} = \frac{a+b}{a-b} $$
Cross-multiplying yields:
$$ (a^2-b^2)(a-b) = (a+b)(a+b) $$
- Simplifying the expression
This expands to:
$$ a^3 - b^3 = a^2 + 2ab + b^2 $$
This requires further simplification based on ( a ) and ( b ).
- Finding the third proportional for ( a^3+b^3, a^2-ab+b^2 ) and ( a-b )
Set up the proportion:
$$ \frac{a^3+b^3}{a^2-ab+b^2} = \frac{a^2-ab+b^2}{a-b} $$
Cross-multiply:
$$ (a^3+b^3)(a-b) = (a^2-ab+b^2)(a^2-ab+b^2) $$
- Conclusion
The calculation leads to a more complex polynomial depending on the supervisory conditions of ( a ) and ( b ).
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For numbers ( 6, 18, 54 ), the third proportional is ( 54 ).
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For ( (x+y)^2, x^3+y^3 ), and ( x+y ), the third proportional depends on specific values of ( x ) and ( y ).
-
For ( a^2-b^2, a+b ), and ( a-b ), and related sums, the third proportional requires calculating expressions respective to ( a ) and ( b ).
-
For ( a^3+b^3, a^2-ab+b^2 ), and ( a-b ), calculations follow similarly.
More Information
The third proportional helps identify consistent relationships in ratios, especially useful in algebra and geometry. This requires understanding how values scale.
Tips
- Misapplying the formula for proportionality.
- Not simplifying expressions properly.
- Forgetting to cross-multiply correctly.
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