Mathbee CBSE Mathbee · CBSE Class 12 Maths
NCERT Class 12 · Ch 6 Miscellaneous Exercise · Q15

NCERT Class 12 Maths · Chapter 6 Applications Of Derivatives · Miscellaneous Exercise Question 15

A cylinder is inscribed coaxially in a right circular cone of height H and semi-vertical angle α. Show its maximum-volume height is H/3 and Vmax=(4/27)πH³tan²α.
Optimal cylinder in cone: representative H=3, tan α=1
Optimal cylinder in cone: representative H=3, tan α=1

💡 Concept Hints

  1. Identify the domain and the inner and outer functions, or the relevant one-sided limits.
  2. Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.

✅ Step-by-Step Solution

Final Answer: h=H/3; Vmax=4πH³tan²α/27.
  1. For cylinder height h from the cone base, similar triangles give cylinder radius r=(H−h)tan α, 0<h<H.
  2. Thus V=πtan²α·h(H−h)². Its derivative is πtan²α(H−h)(H−3h).
  3. The only feasible interior stationary point is h=H/3, with derivative changing + to − and endpoint volumes zero.
  4. At that height r=(2H/3)tan α and Vmax=4πH³tan²α/27.

Master Chapter 6: Applications Of Derivatives

Practise interactive questions, flashcards, visual mind maps, and timed board tests for Applications Of Derivatives.
📘 Chapter Notes & All Exercises 🚀 Open in Interactive Workspace →
← Previous Question
Find the height and volume of the maximum-volume cylinder inscribed…
Next Question →
A cylindrical tank of radius 10 m receives wheat at 314 m³/h. Find…

All Questions in Applications Of Derivatives — Miscellaneous Exercise

Q1Q2Q3Q4Q5Q6Q7Q8Q9Q10Q11Q12Q13Q14Q15Q16

Related Questions in Applications Of Derivatives

Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximal open interval on which f(x)=x²e^(−x) has positive derivative.
Class 12 Applications Of Derivatives · Mcq Single Correct
For f(x)=x²−4x+6, identify the maximal open interval where f′(x)>0.
Class 12 Applications Of Derivatives · Mcq Single Correct
For x>0, at which x is f(x)=x^x stationary?
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximum value of (1/x)^x for x>0.
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximum of [x(x−1)+1]^(1/3) on 0≤x≤1.
Class 12 Applications Of Derivatives · Mcq Single Correct
A trapezium area model is A(x)=(10+x)√(100−x²) cm² for 0≤x≤10. Find its maximum.