Question

R ആരമുള്ള ഒരു ഗോളത്തിന് ഉള്ളിൽ ആലേഖനം ചെയ്യാൻ പറ്റുന്ന പരമാവധി വ്യാപ്തമുള്ള സിലിണ്ടാറിൻ്റെ ഉയരം എത്ര?

• A. $$4R/\sqrt{3}$$
• B. $$2R/\sqrt{3}$$
• C. $$4R^2/\sqrt{3}$$
• D. $$2R^2/3$$

Answer 1

Answer:

$$2R/\sqrt{3}$$

Let 'r' be the radius of the cylinder and 'h' be its height. The radius of the sphere is given as 'R'.

Using the Pythagorean theorem, we have: r² + (h/2)² = R², or r² = R² - (h²/4).
The volume of the cylinder is given by V = πr²h. Substituting the expression for r² we get V = π(R² - h²/4)h = πR²h - (π/4)h³.

To find the maximum volume, we need to find the critical points of the volume equation. This is done by taking the derivative of V with respect to h and setting it to zero: dV/dh = πR² - (3π/4)h² = 0.

Solving for h, we get h² = (4/3)R², or h = 2R/√3.

We can verify that this value of h corresponds to a maximum volume by taking the second derivative of V and checking that it is negative. The second derivative is d²V/dh² = -(3π/2)h, which is negative for the positive value of h we found.

Therefore, the height of the cylinder with maximum volume is 2R/√3.