# The equation that depicts the frequency of a given vibrating string which is under constant tension f and varies inversely with respect to its length l and use it to find out the length the string must be shortened to vibrate with f = 660 if the violin string is 12 inches long when f = 440

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1, Problem 136RE
To determine

## To calculate:The equation that depicts the frequency of a given vibrating string which is under constant tension f and varies inversely with respect to its length l and use it to find out the length the string must be shortened to vibrate with f=660 if the violin string is 12 inches long when f=440

Expert Solution

The string must be shortened to l=8 inches to vibrate with f=660 times per second.

### Explanation of Solution

Given information:

Here, the frequency of a given vibrating string which is under constant tension f and varies inversely with respect to its length l

Also, violin string is 12 inches long when f=440

New frequency is f=660

Formula used:

For 2 variables say, x and y , the statement x is directly proportional to y can be written as:

xαy

Which can be written as:

x=ky

Where k denotes the proportionality constant.

Similarly the statement x is inversely proportional to y can be interpreted as:

xα1y

Which can be written as:

x=k(1y)

Where k denotes the proportionality constant.

Calculation:

As the frequency of a given vibrating string which is under constant tension f and varies inversely with respect to its length l

Recall, For 2 variables say, x and y , the statement x is inversely proportional to y can be interpreted as:

xα1y

Which can be written as:

x=k(1y)

Where k denotes the proportionality constant.

Hence, this variation can be expressed as follows:

fα1l

f=kl (1)

Where k denotes the proportionality constant.

It is also given f=440 times per secwhen d=12 inches

Put these values in (1) to get:

440=k12k=440*12k=5280

Therefore, the proportionality constant k=5280 inches per second.

Now for new frequency of vibrating string f=660 times per second

Replace the value of k=5280 and f=660 in (1) to get:

660=5280ll=5280660l=8

Thus, the string must be shortened to l=8 inches to vibrate with f=660 times per second.

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