Question

Find the semi-vertical angle of the cone of maximum curved surface area that can be inscribed in a given sphere of radius R.

Solution

Correct option is

Let h be the height of cone and be the radius of the cone.

Consider the right ∆OMC where O is the centre of sphere and AM is perpendicular to the base BC of cone. 

          

          

and     r2 + h2 = l2                               …(ii)

Where l is the slant height of cone. 

Curve surface area = C = π r l

Using (i) and (ii), express C in terms of h only.

         

We will maximize C2.

  

         

  

         

         

Hence curved surface area is maximum for   

Using (i), we get:   

          

Semi-vertical angle 

SIMILAR QUESTIONS

Q1

Find points of local maximum and local minimum of f(x) = x2 ex.

Q2

Find points of local maximum and local minimum of (x) = x2/3 (2x – 1). 

Q3

Identify the absolute extrema for the following function.  

                      f (x) = x2 on [–1, 2]

Q4

Identify the absolute extrema for the following function.  

                     f (x) = x3   or    [–2, 2]

Q5

Determine the absolute extrema for the following function and interval.

             g(t) = 2t3 + 3t2 – 12t + 4    on    [– 4, 2]  

Q6

Find the local maximum and local minimum values of the function y = xx.

Q7

A window is in the form of a rectangle surmounted by a semi-circle. The total area of window is fixed. What should be the ratio of the areas of the semi-circular part and the rectangular part so that the total perimeter is minimum?   

Q8

A box of constant volume C is to be twice as long as it is wide. The cost per unit area of the material on the top and four sides is three times the cost for bottom. What are the most economical dimensions of the box?  

Q9

Find the maximum surface area of a cylinder that can be inscribed in a given sphere of radius R 

Q10

Find the point on the curve y = x2 which is closest to the point A(0, a).