The harmonic mean of two numbers, a and b, equals StartStartFraction 2 over StartFraction 1 over a EndFraction plus StartFraction 1 over b EndFraction EndEndFraction . As you vary the length of a violin or guitar string, its pitch changes. If a full-length string is 1 unit long, then many lengths that are simple fractions produce pitches that harmonize, or sound pleasing together. The harmonic mean relates two lengths that produce harmonious sounds. Find the harmonic mean for the string lengths of 1 and one-half .

Question

The harmonic mean of two numbers, a and b, equals StartStartFraction 2 over StartFraction 1 over a EndFraction plus StartFraction 1 over b EndFraction EndEndFraction . As you vary the length of a violin or guitar string, its pitch changes. If a full-length string is 1 unit long, then many lengths that are simple fractions produce pitches that harmonize, or sound pleasing together. The harmonic mean relates two lengths that produce harmonious sounds. Find the harmonic mean for the string lengths of 1 and one-half .

(1 point)
Responses

StartFraction 3 over 2 EndFraction
Image with alt text: StartFraction 3 over 2 EndFraction

StartFraction 2 over 3 EndFraction
Image with alt text: StartFraction 2 over 3 EndFraction

StartFraction 2 over 5 EndFraction
Image with alt text: StartFraction 2 over 5 EndFraction

1

Answer 1

The harmonic mean of two numbers, a and b, is given by the formula:

Harmonic mean = 2 / ((1/a) + (1/b))

In this case, a = 1 and b = 1/2. Plugging these values into the formula, we get:

Harmonic mean = 2 / ((1/1) + (1/(1/2)))
= 2 / (1 + 2)
= 2 / 3

Therefore, the harmonic mean for the string lengths of 1 and one-half is 2/3.