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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

The speeds of sound c 1 in an upper layer and c 2 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 T 1 seconds. The next signal travels from P to a point R, from R to S in the lower layer, and then to Q, taking T 2 seconds. The third signal is reflected off the lower layer at the midpoint O of RS and takes T 1 seconds to reach Q. (See the figure.)

Chapter 3.P, Problem 19P, The speeds of sound c1 in an upper layer and c2 in a lower layer of rock and the thickness h of the

(a) Express T 1 T 2 , and T 3 in terms of D , h , c 1 , c 2 , and θ .

(b) Show that T 2 is a minimum when sin θ = c 1 / c 2

(c) Suppose that D = 1 k m , T 1 = 0.26 s , T 2 = 0.32 s , and T 3 = 0.34 s Find c 1 , c 2 and h .

Note: Geophysicists use this technique when studying the structure of the earth’s crust, whether searching for oil or examining fault lines.

To determine

a)

To express:

T1,T2  and T3 in terms of D, h,c1,c2 and θ

Explanation

1) Concept:

The function is minimum when f'x<0  and maximum when f'x>0.

Power rule of differentiation ddxxn=nxn-1

Chain rule ddxfxgx=fxddxgx+gxddxf(x)

time=distancespeed

2) Calculation:

time=distancespeed

T1=Dc1

T2=2(PR)c1+RSc

To determine

b)

To show:

T2 is minimum when sin θ=c1c2

To determine

c)

To find:

The values of h,c1,c2 when T1=026s, T2=0.325s,  T3=0.34s, D=1km.

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