Show that the volume of greatest culinder that can be inscribed in a cone of height h and semi vertical angle alpha is 4/27 pi h cube tan square alpha
Here is the answer to your query.
Let VAB be a given cone of height h, semi-vertical angle α and let x be the radius of the base of the cylinder A´ B´ DC. Which is inscribed in the cone VAB.
OO´ = height of the cylinder
⇒ OO´ = VO – VO´ = h – x cot α
Let V be the volume of the cylinder. Then
V = πx2 (h – x cot α) ...... (1)
for maximum of minimum V we must have
Hence V is maximum when
The maximum volume of the cylinder is given by
in all maxima and minima word problems u haev to represent the equation in one variable like in this question,the hieght and the radius of the cylinder are the variables and since the height of the cylinder is given threfore represent the radius of the cylinder in terms of h and find the maxima or minima