CBSE Class 12 Mathematics 2026 Main Paper 65(B) · Q335 Marksoptimization
CBSE Class 12 Mathematics 2026 Main Paper 65(B) · Question 33 (Applications Of Derivatives)
Find dimensions of the closed circular cylinder of volume 200 cm³ with minimum surface area. OR Show that a cone of maximum volume inscribed in a sphere of radius r has height 4r/3.
Alternative 1 5 Marks
Find dimensions of the closed circular cylinder of volume 200 cm³ with minimum surface area.
Answer: r=(100/π)^(1/3) cm and h=2r.
- Eliminate h=200/(πr²) to get area S=2πr²+400/r.
- Set S′=4πr−400/r²=0, giving r³=100/π.
- Since S″=4π+800/r³>0, this is the minimum and the volume constraint gives h=2r.
Alternative 2 5 Marks
Show that a cone of maximum volume inscribed in a sphere of radius r has height 4r/3.
Answer: h=4r/3.
- For height h, the base radius squared is r²−(r−h)²=2rh−h².
- The volume is (π/3)h²(2r−h) on 0≤h≤2r. Its derivative is (π/3)h(4r−3h).
- It increases up to 4r/3 and decreases afterward, proving the maximum.