A thin string is held at one end and oscillates vertically when driven by a motor so that \[y(x=0,t)=8\sin4t\si{\ \centi\meter}\] The string's linear mass density is \(0.2\text{ kg m}^{-1}\), its tension is \(1\text{ N}\), and its length is \(\si{10\ \centi\meter}\).

Suppose the string is now driven by the same motor inside a bath filled with \(\si{1\ \kilo\gram}\) water. Due to friction heat is transferred to the bath with a heat transfer efficiency of 50%. Calculate how much time (in seconds) passes before the temperature of the bath rises by \(\si{1\ \celsius}\).

**Details and Assumptions:**

- Specific heat of water = \(\si{4.2\ \kilo\joule/\kilo\gram\per\kelvin}\)
- Neglect the effect of gravity.

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