Answer
🎯 Objective
Determine the depth of a tumor inside the liver using Snell’s Law and trigonometric relationships involving ultrasonic waves.
📘 Concept Used – Snell’s Law
Snell’s Law equation:
sin(θ₁) / v₁ = sin(θ₂) / v₂
- θ₁: Incident angle in surrounding medium (55°)
- θ₂: Refracted angle in liver
- v₁: Wave speed in surrounding medium
- v₂: Wave speed in liver = 0.941 × v₁
- y₁: Vertical distance = 14.0 cm
🔢 Step 1 – Apply Snell’s Law
Calculate sin(θ₂):
sin(θ₂) = 0.941 × sin(55°) = 0.941 × 0.8192 = 0.771
θ₂ ≈ sin⁻¹(0.771) ≈ 50.5°
📐 Step 2 – Horizontal Distance (x)
Using tan(θ₁) to find horizontal distance:
x = y₁ × tan(θ₁) = 14.0 × tan(55°) = 14.0 × 1.428 = 19.99 cm
📏 Step 3 – Depth in Liver (y₂)
Using tan(θ₂):
y₂ = x / tan(θ₂) = 19.99 / tan(50.5°) = 19.99 / 1.209 ≈ 16.54 cm