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Class 12 Applications Of Derivatives · Very Long Answer5 Marksrelated_rates

Class 12 Maths · Applications Of Derivatives · Very Long Answer #61

The median of an equilateral triangle increases at 2√3 cm/s. Find the rate at which its side increases.
Equilateral triangle and its median
Equilateral triangle and its median

✅ Step-by-Step Solution

Final Answer: 4 cm/s
  1. Let its side be a and median length m. In an equilateral triangle the median is also an altitude.
  2. It bisects the base, so m²=a²−(a/2)²=3a²/4.
  3. Positive lengths give m=√3a/2.
  4. Differentiating gives ṁ=(√3/2)ȧ; insert ṁ=2√3.
  5. Thus ȧ=2√3·2/√3=4 cm/s.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Let its side be a and median length m. In an equilateral triangle the median is also an altitude.
  • [1 mark] It bisects the base, so m²=a²−(a/2)²=3a²/4.
  • [1 mark] Positive lengths give m=√3a/2.
  • [1 mark] Differentiating gives ṁ=(√3/2)ȧ; insert ṁ=2√3.
  • [1 mark] Thus ȧ=2√3·2/√3=4 cm/s.

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