Questions: The speed v (λ/10 m/sec) of an ocean wave in deep water is approximated by v(λ) = 4√λ, where λ (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 = □ m/sec per meter.

Solution Steps

To find the average rate of change of the speed of the wave between wavelengths of 1 meter and 4 meters, we need to calculate the difference in speed at these two wavelengths and divide it by the difference in the wavelengths. This is essentially finding the slope of the secant line between the two points on the curve defined by the function \( v(\lambda) = 4\sqrt{\lambda} \).

Using the function \( v(\lambda) = 4\sqrt{\lambda} \), we find the speed at \( \lambda = 1 \) m: \[ v(1) = 4\sqrt{1} = 4.0 \, \text{m/sec} \]

Next, we calculate the speed at \( \lambda = 4 \) m: \[ v(4) = 4\sqrt{4} = 4 \times 2 = 8.0 \, \text{m/sec} \]

The average rate of change in speed between the wavelengths \( \lambda = 1 \) m and \( \lambda = 4 \) m is given by: \[ \text{Average Rate of Change} = \frac{v(4) - v(1)}{4 - 1} = \frac{8.0 - 4.0}{3} = \frac{4.0}{3} \approx 1.3333 \, \text{m/sec per meter} \]

Final Answer