Question 13 of 23beginner🔧 ApplyNumerical2 marks

A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the wavelength of sound in the tissue in which the speed of sound is 1.7 km s−1^{-1}? The operating frequency of the scanner is 4.2 MHz.

Correct Answer

The wavelength of sound in the tissue is approximately 4×104 \times 10⁻⁴ m or 0.4 mm.

Given: Speed of sound in tissue, v = 1.7 km s⁻¹ = 1.7×101.7 \times 10³ m s⁻¹ Operating frequency, ν = 4.2 MHz = 4.2×104.2 \times 10⁶ Hz

Formula: The relationship between speed, wavelength and frequency is: v = λν Therefore, λ = v/ν

Calculation: λ = v/ν λ = (1.7×101.7 \times 10³)/(4.2×104.2 \times 10⁶) λ = (1.7/4.2) × 10⁻³ λ ≈\approx 0.405×100.405 \times 10⁻³ m λ ≈\approx 4.05×104.05 \times 10⁻⁴ m ≈\approx 0.4 mm

Exercise: EXERCISES | Q: 14.7 | (Page 21)
For More Understanding

Explanation

This question tests the fundamental relationship between wave speed, wavelength, and frequency. From the context, the formula λ = v/ν (Equation 14.15) establishes that wavelength equals speed divided by frequency. The key challenge is unit conversion: converting 1.7 km s⁻¹ to 1.7×101.7 \times 10³ m s⁻¹ and 4.2 MHz to 4.2×104.2 \times 10⁶ Hz before applying the formula.

Solution Steps

  1. Step 1: Convert speed to SI units: v = 1.7 km s⁻¹ = 1.7×101.7 \times 10³ m s⁻¹ = 1700 m s⁻¹

  2. Step 2: Convert frequency to SI units: ν = 4.2 MHz = 4.2×104.2 \times 10⁶ Hz

  3. Step 3: Apply the wavelength formula: λ = v/ν

  4. Step 4: Calculate: λ = (1.7×101.7 \times 10³)/(4.2×104.2 \times 10⁶) = (1.7/4.2) × 10⁻³ ≈\approx 4.05×104.05 \times 10⁻⁴ m ≈\approx 0.4 mm

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