CBSE Class 12 Mathematics 2026 Main Paper 65/4/2 · Q374 Marksrelated_rates
CBSE Class 12 Mathematics 2026 Main Paper 65/4/2 · Question 37 (Applications Of Derivatives)
Orange juice fills a conical cup of depth 15 cm and top radius 5 cm at 0.1 cm³/s. The juice has depth h and surface radius r. The cone’s semi-vertical angle is α.
Part (1) 1 Marks
Relate r and h using α.
Answer: r=h tan α=h/3.
- From the cross section, tan α=r/h.
- The cup gives tan α=5/15=1/3.
Part (2) 1 Marks
At what rate does the level rise at h=6?
Answer: 1/(40π) cm/s.
- Use V=πh³/27 and differentiate to get dV/dt=(πh²/9) dh/dt.
- Set h=6 and dV/dt=1/10 to obtain dh/dt=1/(40π).
Part (3) 2 Marks
At h=6, find the rate of increase of the top surface area.
Answer: 1/30 cm²/s.
- The top area is A=πr²=πh²/9.
- Differentiate to get dA/dt=(2πh/9)dh/dt.
- At h=6, substitute dh/dt=1/(40π) to obtain 1/30.
Part (4) 2 Marks
At h=6, find the rate of increase of the wetted curved surface area.
Answer: √10/30 cm²/s.
- The slant length is √(r²+h²)=√10·h/3.
- The wetted curved area is πr times the slant length, so S=√10·πh²/9.
- Its rate is √10 times the top area rate, hence √10/30.