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Chapter 14: Problem 21

Medical ultrasound waves travel at about \(1500 \mathrm{m} / \mathrm{s}\) in soft tissue. Higher frequencies provide clearer images but don't penetrate to deeper organs. Find the wavelengths of (a) 8.0 -MHz ultrasound used in fetal imaging and (b) 3.5 -MHz ultrasound used to image an adult's kidneys.

Short Answer

Expert verified
The wavelength of the 8.0-MHz ultrasound used in fetal imaging is approximately \(0.1875\) mm. The wavelength of the 3.5-MHz ultrasound used in kidney imaging is approximately \(0.4286\) mm.

Step by step solution

01

Applying the Wave Formula for Fetal Imaging

We start by substituting the given values for speed and frequency into the wave equation to find the wavelength for the 8.0-MHz ultrasound used in fetal imaging. Note that 1 MHz = \(10^6\) Hz. So the frequency \(f = 8.0 \times 10^6\) Hz. The wave formula becomes: \(1500 = 8.0 \times 10^6 \times \lambda\)
02

Solve for Wavelength in Fetal Imaging

Rearranging to solve for wavelength, the equation becomes: \(\lambda = \frac{1500}{8.0 \times 10^6}\). Calculating this gives us approximately \(1.875 \times 10^{-4}\) meters or 0.1875 mm.
03

Applying the Wave Formula for Kidney Imaging

Next we substitute speed and frequency values into the wave equation for the 3.5-MHz ultrasound used for kidney imaging. The frequency here is \(f = 3.5 \times 10^6\) Hz. So the wave equation here is: \(1500 = 3.5 \times 10^6 \times \lambda\).
04

Solve for Wavelength in Kidney Imaging

Rearranging to solve for λ gives us \(\lambda = \frac{1500}{3.5 \times 10^6}\). Performing this calculation, we get approximately \(4.286 \times 10^{-4}\) meters or 0.4286 mm.

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