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# The speeds of sound c1 in an upper layer and c2 in a lower ISBN: 9781285740621 127

## Solution for problem 21 Chapter 4

Calculus | 8th Edition

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Problem 21

The speeds of sound c1 in an upper layer and c2 in a lower layer of rock and the thickness h of the upper layer can be determined by seismic exploration if the speed of sound in the lower layer is greater than the speed in the upper layer. A dynamite charge is detonated at a point P and the transmitted signals are recorded at a point Q, which is a distance D from P. The first signal to arrive at Q travels along the surface and takes T1 seconds. The next signal travels from P to a point R, from R to S in the lower layer, and then to Q, taking T2 seconds. The third signal is reflected off the lower layer at the midpoint O of RS and takes T3 seconds to reach Q. (See the figure.) (a) Express T1, T2, and T3 in terms of D, h, c1, c2, and . (b) Show that T2 is a minimum when sin c1yc2. (c) Suppose that D 1 km, T1 0.26 s, T2 0.32 s, and T3 0.34 s. Find c1, c2, and h. Note: Geophysicists use this technique when studying the structure of the earths crust, whether searching for oil or examining fault lines.

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##### ISBN: 9781285740621

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