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Chapter 12: Problem 61

According to a treasure map, a treasure lies at a depth of 40.0 fathoms on the ocean floor due east from the lighthouse. The treasure hunters use sonar to find where the depth is 40.0 fathoms as they head cast from the lighthouse. What is the elapsed time between an emitted pulse and the return of its echo at the correct depth if the water temperature is $\left.25^{\circ} \mathrm{C} ? \text { [Hint: One fathom is } 1.83 \mathrm{m} .\right]$

Short Answer

Expert verified
Answer: The elapsed time between the emitted pulse and the return of its echo is approximately 0.0975 seconds.

Step by step solution

01

Convert depth to meters

Given depth is 40 fathoms. The conversion for fathoms to meters is 1 fathom = 1.83 meters. Depth in meters = Depth in fathoms × Conversion factor Depth in meters = 40 fathoms × 1.83 meters/fathom Depth in meters = 73.2 meters
02

Find the speed of sound in water

We can use the empirical formula for the speed of sound in water, which indicates that the speed of sound (v) in meters per second (m/s) is given by \(v = 1449.2 + 4.6 T - 0.055 T^2 + 0.00029 T^3\) where \(T\) is the temperature in Celsius. For the given temperature of 25°C, we can plug it into the formula to get the speed of sound. \(v = 1449.2 + 4.6(25) - 0.055(25)^2 + 0.00029(25)^3\) \(v \approx 1501.37\) m/s The speed of sound in water at 25°C is approximately 1501.37 m/s.
03

Calculate the round-trip distance

The sonar pulse travels down to the ocean floor and back up. Therefore, the total round-trip distance will be twice the depth. Round-trip distance = 2 × Depth Round-trip distance = 2 × 73.2 meters Round-trip distance = 146.4 meters
04

Calculate the elapsed time

We can now determine the elapsed time between the emitted pulse and the return of its echo using the formula: Elapsed time = \(\frac{\text{Round-trip distance}}{\text{Speed of sound in water}}\) Elapsed time = \(\frac{146.4 \text{ meters}}{1501.37 \text{ m/s}}\) Elapsed time \(\approx 0.0975\) seconds The elapsed time between the emitted pulse and the return of its echo is approximately 0.0975 seconds.

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