A standing wave produced on a string fixed at both ends has the...

The equation of the standing wave is given by $y = A \sin(kx) \cos(\omega t)$.

Comparing the given equation with the standard form, we get $k = 0.314 \text{ cm}^{-1}$.

We know that $k = \dfrac{2\pi}{\lambda}$.

$\dfrac{2\pi}{\lambda} = 0.314 \Rightarrow \dfrac{2 \times 3.14}{\lambda} = 0.314 \Rightarrow \lambda = 20 \text{ cm}$.

For a string fixed at both ends, the length of the string must be an integral multiple of $\dfrac{\lambda}{2}$.

$L = n\dfrac{\lambda}{2}$ where $n = 1, 2, 3, \ldots$

For the smallest possible length, $n = 1$.

L_\min = \dfrac{\lambda}{2} = \dfrac{20}{2} = 10 \text{ cm}.