On a string instrument of the violin family, the effective length of a string is the distance between the bridge and the nut.

For a violin, this distance is 31.1 cm, while for a cello it is 38.8 cm.

The string of a violin is placed in a cello with the intention of producing a sound of the same fundamental frequency.

To accomplish this, the string on the cello will be under a larger tension than on the violin.

By how much should the tension in the cello be increased with respect to the tension in the violin?

Express the result as a percentage, and to two significant figures. Only answer in numerical values, without the % sign.

For example, an increase of 11% corresponds to Tcello = (1.11) Tviolin, and should be entered as 11 in the answer box.

To determine the tension in the cello string relative to the tension in the violin string, we need to consider the relationship between tension, frequency, and the effective length of the string.

The frequency of a vibrating string is inversely proportional to the square root of the effective length and directly proportional to the square root of the tension. Mathematically, this can be expressed as:

f ∝ √(T/L)

where f is the frequency, T is the tension, and L is the effective length of the string.

Since we want to achieve the same fundamental frequency, we can set up an equation using the proportionality:

f(cello) = f(violin)

√(T(cello)/L(cello)) = √(T(violin)/L(violin))

Taking the square of both sides to eliminate the square root:

T(cello)/L(cello) = T(violin)/L(violin)

Rearranging the equation, we get:

T(cello) = (L(cello)/L(violin)) * T(violin)

Now, let's substitute the given values into the equation:

T(cello) = (38.8 cm / 31.1 cm) * T(violin)

Simplifying the expression, we find:

T(cello) = 1.247 * T(violin)

To express the tension in the cello relative to the tension in the violin as a percentage, we calculate the increase in tension:

Increase in tension = (T(cello) - T(violin)) / T(violin) * 100

Substituting the values, we get:

Increase in tension = (1.247 * T(violin) - T(violin)) / T(violin) * 100

Simplifying the expression, we find:

Increase in tension = (0.247 * T(violin)) / T(violin) * 100

The T(violin) on the numerator and denominator cancels out, leaving us with:

Increase in tension = 0.247 * 100

Finally, calculating the percentage increase, we find that the tension in the cello should be increased by 24.7% with respect to the tension in the violin.

Therefore, the answer to the question is 24.7.