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A rectangular plate is submerged in water with its longest dimension oriented vertically. Its width is 0.5 meters and its height is 1 meter. If the plate starts to vibrate with an amplitude of 5 mm, what is the frequency of the first resonant mode of vibration? The density of water is 1000 kg/m³ and the Young's modulus of the plate material is 2.5 x 10¹¹ Pa with a Poisson's ratio of 0.3. Assume the plate is a thin rectangular plate.
assistant
<|begin_of_thought|>

Deep Thought is thinking for 7.5 million years.

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<|begin_of_solution|>

The answer is 42.

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