Question
The speed \( \nu(L) \) (in \( \mathrm{m} / \mathrm{sec} \) ) of an ocean wave in deep water is approximated by \( v(L)=0.6 \sqrt{L} \), where \( L \) (in meters) is the wavelength of the wave. (The wavelength is the distance between two consecutive wave crests.) Part: \( 0 / 4 \) Part 1 of 4 (a) Find the average rate of change in speed between waves that are between 1 m and 4 m in length. The average rate of change in speed between waves that are between 1 m and 4 m in length is
Ask by Curry Lindsey.
Mar 04,2025 23:41
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The average rate of change in speed between waves that are between 1 meter and 4 meters in length is 0.2 meters per second per meter.
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To find the average rate of change in speed \( \nu(L) \) between wavelengths of 1 m and 4 m, we first calculate the speeds at these two wavelengths using the formula \( \nu(L) = 0.6 \sqrt{L} \).
1. For \( L = 1 \) m:
\[
\nu(1) = 0.6 \sqrt{1} = 0.6 \text{ m/sec}
\]
2. For \( L = 4 \) m:
\[
\nu(4) = 0.6 \sqrt{4} = 0.6 \times 2 = 1.2 \text{ m/sec}
\]
Next, we calculate the average rate of change in speed:
\[
\text{Average Rate of Change} = \frac{\nu(4) - \nu(1)}{4 - 1} = \frac{1.2 - 0.6}{3} = \frac{0.6}{3} = 0.2 \text{ m/sec/m}
\]
So, the average rate of change in speed between waves that are between 1 m and 4 m in length is \( 0.2 \text{ m/sec/m} \).
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