A guitar string is supposed to be played with a fundamental frequency of 880 Hz. If it is determined to have, instead, a fundamental frequency of 882 Hz, by what amount (as a percentage) must the tension in the string be changed in order that the fundamental frequency be restored to 880 Hz?

The frequency of a vibrating string is given by the formula:

f = (1/2L) * sqrt(T/μ)

where: f is the frequency, L is the length of the string, T is the tension in the string, and μ is the linear density of the string.

We can see from this formula that the frequency is proportional to the square root of the tension. Therefore, if we want to find out by what percentage the tension must be changed to restore the fundamental frequency to 880 Hz, we can set up the following equation:

sqrt(T1/T2) = f1/f2

where: T1 is the initial tension (which we are trying to find), T2 is the final tension, f1 is the initial frequency (882 Hz), and f2 is the final frequency (880 Hz).

Solving this equation for T1 gives us:

T1 = T2 * (f1/f2)^2

We can then find the percentage change in tension by subtracting T2 from T1, dividing by T2, and multiplying by 100:

% change = ((T1 - T2) / T2) * 100

Substituting the expression for T1 from above gives us:

% change = ((T2 * (f1/f2)^2 - T2) / T2) * 100 = ((f1/f2)^2 - 1) * 100

Substituting the given values for f1 and f2 gives us:

% change = ((882/880)^2 - 1) * 100 = (1.00454545^2 - 1) * 100 = (1.00913242 - 1) * 100 = 0.913242%

So, the tension in the string must be increased by approximately 0.913% in order for the fundamental frequency to be restored to 880 Hz.