When the narrow beam of ultrasound waves reflects off a liver tumor, the depth of the tumor is found to be 6.30 cm.
Snell's law states that the ratio of the sin of the angle of incidence to the angle of reflection is equal to the ratio of the speed of the wave in the first medium to the speed of the wave in the second medium.
[tex]\dfrac{sin\:i}{sin\:r} =\dfrac{v_{1} }{v_{2} }[/tex]
Where, i represents the angle of incidence and r is the angle of reflection.
Given the speed of sound in the liver is 10% less than in the medium, so
[tex]\dfrac{v_{liver} }{v_{medium} }=0.900[/tex]
According to the figure, the angle of incidence is 50°, then from the Snell's law, we have
[tex]sin\:r=(\dfrac{v_{liver} }{v_{medium} } )\times sin\:i\\\\r=sin^{-1} [(0.900)sin 50][/tex]
The angle of reflection comes out to be
r = 43.6°
From the law of reflection,
[tex]d =\dfrac{12\:\rm cm}{2} =6\:\rm cm[/tex]
The depth of the tumor is
[tex]h=\dfrac{d}{tan\:r} =\dfrac{6}{tan\:43.6} \\\\h=6.30\:\rm cm[/tex]
The depth of the tumor in the liver is 6.30 cm.
Learn more about Snell's law.
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