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If the frequency f varies directly with the square root of the tension T and inversely with the length L, it could be expressed by the formula
`f = ksqrt(T)/L` , where k is the proportionality constant, a number independent on T and L.
If the length is doubled, it will become 2L, and if the tension is quadrupled, it will become 4T. The constant k will remain the same, as it does not depend on L and T.
The new frequency, F, will then become
`F = k sqrt(4T)/(2L)`
The square root of 4T can be simplified as follows:
`sqrt(4T) = sqrt(4)* sqrt(T) = 2sqrt(T)`
Plugging this into the expression for F above, we get
`F = k (2sqrt(T))/(2L) = k sqrt(T)/L`
Comparing the final result with the original formula for frequency f, we see that they are the same. So, F = f, which means that
the frequency will not change if the tension is quadrupled and the length is doubled.