Math/Music/Hz Word Problem

Question

An acoustic guitar’s (lower) E-string is 65 centimeters long from bridge to nut and vibrates at a frequency of 165 Hz (E3). When that string is pressed down just behind the third fret, it vibrates between that fret and the bridge at a frequency of 196.22 Hz (G3). When that string is pressed down just behind the sixth fret, it vibrates between that fret and the bridge at a frequency of 233.35 Hz (B_3^b). Find the distance between the third fret and the sixth fret on this guitar. Round your answer to the nearest 0.1 centimeter Hint: there is no shortcut. First find the distance from the bridge to the third fret…

Answer 1

We'll use proportions for this problem again. This time, string length is inversely proportional to frequency. So, frequency goes up as string length goes down. That means we should use the multiplicative inverse of the given frequencies in our proportions. The inverse of 165 is just 1/165, the inverse of 196.22 is 1/196.22, and the inverse of 233.35 is 1/233.35.

 

 

Let's find the distance from the bridge to the third fret:

 

(string length) / (inverse of frequency) = (smaller string length) / (inverse of smaller string's frequency)

65 / (1/165) = x / (1/196.22)

65(1/196.22) = (1/165)x

65/196.22 = x/165

10725 = 196.22x        (I cross-multiplied again)

54.66 = x

 

The distance from the bridge to the third fret is approximately 54.66 cm.

 

 

Now, let's do the same to find the distance from the bridge to the sixth fret:

 

65 / (1/165) = x / (1/233.35)

65/233.35 = x/165

10725 = 233.35x        (I cross-multiplied again)

45.96 = x

 

The distance from the bridge to the sixth fret is approximately 45.96 cm.

 

 

Subtract the distance from the bridge to the third fret from the distance from the bridge to the sixth fret to find the distance from the third fret to the sixth fret:

 

54.66 cm - 45.96 cm = 8.7 cm between the third fret and the sixth fret