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How Do You Solve a Word Problem Using Inverse Variation?

The pitch of a musical tone varies inversely with its wavelength. A tone that has a pitch of 352 vibrations per second has a wavelength of 3 feet. Find the wavelength of a pitch with 88 vibrations per second.

Summary

  1. p represents pitch
  2. w represents wavelength
  3. pw=k is the formula for inverse variation
  4. k is the constant of inverse variation
  5. Plug 352 and 3 into the formula to find k, which is 1056
  6. 88 is our value for p, pitch
  7. w is our unknown variable
  8. 1056 is our constant, k
  9. Divide both sides by 88 to solve for w

Notes

    1. 'Varying inversely' means 'inverse variation'
    1. The formula for inverse variation is xy=k, where x and y are variables and k is a constant
    1. The formula for inverse variation is xy=k, where x and y are variables and k is a constant
    1. The formula for inverse variation is xy=k, where x and y are variables and k is a constant
    1. The formula for inverse variation is xy=k, where x and y are variables and k is a constant
    2. Since pitch varies inversely with wavelength, pitch and wavelength are the variables in this problem
    1. Since pitch varies inversely with wavelength, pitch and wavelength are the variables in this problem
    1. p will be the variable we use to represent pitch
    2. w will be the variable we use to represent wavelength
    1. p is a variable that represents pitch
    2. w is a variable that represents wavelength
    3. k is the constant of inverse variation
    1. p is a variable that represents pitch, which we know equals 88
    2. w is a variable that represents wavelength
    1. Right now we have two unknowns, w and k
    2. But we can use the other information in our problem to find k
    3. Since k is a constant, it won't change even if the variables change
    1. We can use the other information in our problem to find k
    2. Since k is a constant, it won't change even if the variables change
    1. Since p represents pitch, p here is 352
    2. Since w represents wavelength, w here is 3
    1. We can plug 352 and 3 into the formula for inverse variation to find k
    2. Since k is a constant, it will stay the same even if the variables change
    1. Plug 352 in for p and 3 in for w
    1. 352•3=1056
    2. So k=1056
    1. Now that we know that k=1056, we can use find the wavelength in our original problem
    1. Remember, w was the variable we defined to represent wavelength
    1. Plug 88 in for p and 1056 in for k in the formula pw=k
    2. So we have 88w=1056
    1. Division is the opposite of multiplication
    2. On the left the 88's cancel, leaving us with w
    3. On the right, 1056/88=12
    4. So w, the wavelength, equals 12