SOLUTION: if arithmetic mean and geometric mean are inserted between a and b, such that arithmetic mean is double the geometric mean . show that ratio of a and b is [2+(3)^(1/2)]/[2-(3)^(1/

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Question 973163: if arithmetic mean and geometric mean are inserted between a and b, such that arithmetic mean is double the geometric mean . show that ratio of a and b is [2+(3)^(1/2)]/[2-(3)^(1/2)]
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
If an arithmetic mean and a geometric mean are inserted between a and b, such
that the arithmetic mean is double the geometric mean. show that ratio of a and
b is [2+(3)^(1/2)]/[2-(3)^(1/2)]
We change the 1%2F2 powers to square roots
 
%282%2Bsqrt%283%29%29%2F%282-sqrt%283%29%29

Then we rationalize the denominator:

%28%282%2Bsqrt%283%29%29%29%2F%28%282-sqrt%283%29%29%29%22%D7%22%28%282%2Bsqrt%283%29%29%29%2F%28%282%2Bsqrt%283%29%29%29

%284%2B4sqrt%283%29%2B3%29%2F%284-3%29

%287%2B4sqrt%283%29%29%2F1

7%2B4sqrt%283%29

matrix%281%2C4%2CArithmetic%2Cmean%2C%22%22=%22%22%2C%28a%2Bb%29%2F2%29

matrix%281%2C4%2CGeometric%2Cmean%2C%22%22=%22%22%2Csqrt%28ab%29%29

>>...the arithmetic mean is double the geometric mean...<<
%28a%2Bb%29%2F2%22%22=%22%222sqrt%28ab%29

Multiply both sides by 2 to clear the fraction:

a%2Bb%22%22=%22%224sqrt%28ab%29

Square both sides:

%28a%2Bb%29%5E2%22%22=%22%22%284sqrt%28ab%29%29%5E2

a%5E2%2B2ab%2Bb%5E2%22%22=%22%2216ab

a%5E2-14ab%2Bb%5E2%22%22=%22%22%220%22

a%5E2%2B%28-14b%29a%2Bb%5E2%22%22=%22%22%220%22

Solve for "a" using the quadratic formula.  We will use capital
letters in the quadratic formula to avoid conflict of notation:

a%22%22=%22%22%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29+

where A=1, B=-14b, C=b%5E2

a%22%22=%22%22%28-%28-14b%29+%2B-+sqrt%28+%28-14b%29%5E2-4%281%29%28b%5E2%29+%29%29%2F%282%281%29%29+

a%22%22=%22%22%2814b+%2B-+sqrt%28196b%5E2-4b%5E2+%29%29%2F2+

a%22%22=%22%22%2814b+%2B-+sqrt%28192b%5E2+%29%29%2F2+

a%22%22=%22%22%2814b+%2B-+sqrt%2864%2A3%2Ab%5E2%29%29%2F2+

a%22%22=%22%22%2814b+%2B-+8b%2Asqrt%283%29%29%2F2+

a%22%22=%22%22%282b%287+%2B-+4sqrt%283%29%29%29%2F2+



a%22%22=%22%22%28cross%282%29b%287+%2B-+4sqrt%283%29%29%29%2Fcross%282%29+

a%22%22=%22%22b%287+%2B-+4%2Asqrt%283%29%29+

Divide both sides by b

a%2Fb%22%22=%22%22b%287+%2B-+4%2Asqrt%283%29%29%2Fb+

a%2Fb%22%22=%22%22cross%28b%29%287+%2B-+4%2Asqrt%283%29%29%2Fcross%28b%29+

a%2Fb%22%22=%22%227+%2B-+4%2Asqrt%283%29+

We have shown that the ratio is either 7+%2B+4%2Asqrt%283%29+ or 7+-+4%2Asqrt%283%29+.

So you might point out to your teacher that the problem as it is stated here,
is not necessarily true.

The problem should be stated this way:

If an arithmetic mean and a geometric mean are inserted between a and b, such
that the arithmetic mean is double the geometric mean. show that ratio of a and
b is [2+(3)^(1/2)]/[2-(3)^(1/2)] OR [2-(3)^(1/2)]/[2+(3)^(1/2)].
The second one, when the 1%2F2 powers are changed to square roots and its
denominator is rationalized, becomes 7+-+4%2Asqrt%283%29+.

Edwin