CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Question 22 (Applications Of Derivatives)

Question

A room freshener bottle in the shape of an inverted cone sprays the perfume at regular intervals such that volume of the perfume in the bottle decreases at the steady rate of 1 mm³/min. Find the rate at which level of perfume is dropping at an instant when level of perfume in the bottle is 10 mm, if the semi-vertical angle of conical bottle is π/6.

Accepted Answer

Final Answer: Rate = −3/(100π) mm/min (dropping). Step 1: Let v, r, h be volume, radius, height of perfume. dv/dt = −1. Step 2: Given semi-vertical angle π/6 ⇒ tan(π/6) = r/h = 1/√3 ⇒ r = h/√3. v = (1/3)πr²h = (1/3)π (h²/3)h = (π/9) h³. dv/dt = (π/9)·3h² dh/dt = (π/3)h² dh/dt. −1 = (π/3) h² dh/dt ⇒ dh/dt = −3/(π h²). At h=10, dh/dt = −3/(100π) mm/min. Step 3: Rate = −3/(100π) mm/min (dropping).

CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Question 22 (Applications Of Derivatives)

✅ Step-by-Step Solution

📋 CBSE Stepwise Marking Scheme

Master Chapter 6: Applications Of Derivatives

All Questions in CBSE Class 12 Mathematics 2026 Main Paper 65/1/1

Related Questions in Applications Of Derivatives