Manipal MET2016MathematicsApplication of Derivatives

If is the semi-vertical angle of a cone of maximum volume and given slant height, then is given by

Options

Correct answer

C. 2

Step-by-step solution

Volume of cone, V= 3 r^2 h V= 3 r^2 1^2-r^2 On differentiating w.r.t. I , we get d V d r = 3 [2 r 1^2-r^2 + r^2 2 1^2-r^2 (-2 r) ] Put d V d r =0 , we get r= 1 2 3 At r=1 2 3 , d^2 V d r^2 0, (maximum) h= I^2- 2 3 l^2 = 1 3 In A B C, = r h = 1 2 3 1 / 3 = 2

Practice Application of Derivatives on Quantrex Academy →

More from Application of Derivatives

Consider the quadratic equation a x^2+b x+c=0 , where 2 a+3 b+6 c=0 and let g(x)= a x^3 3 + b x^2 2 +c x . Statement-I : The given quadratic equation ax ^2+ bx + c =0 has at least 2025 The difference between the absolute maximum and absolute minimum values of the function f(x)=2 x^3-15 x^2+36 x-30 on [-1,4] is 2025 If f(x)=x e^ x(1-x) , x R , then f(x) is 2025 The angle between the curves y ^2= x and x ^2= y at the point (1,1) is 2025 If the tangent of the curve 4 y^3=3 a x^2+x^3 drawn at the point (a, a) forms a triangle of area 25 24 sq.units with the coordinate axes then a = 2025 If the function f(x)= x- ^2 x is defined on the interval [- , ] , then f is strictly increasing in the interval 2025 If the Lagrange's mean value theorem is applied to the function f(x)=e^x defined on the interval [1,2] and the value of c (1,2) is k , then e^ k-1 = 2025 If the tangent to the curve x y+a x+b y=0 at (1,1) makes an angle Tan ⁻¹ 2 with X -axis, then ab a + b = 2025 Full Application of Derivatives list All Manipal MET PYQs