Show that the semi vertical angle of a cone of given slant height is tan - 1 2 when its volume is maximum.

Let θ be the semi-vertical angle of the cone. It is clear that θ ∈ 0 , π 2 . Let r , h and l be the radius, height and slant height of the cone respectively. The slant height of the cone is given as constant. Now, r = l sin θ and h = l cos θ The volume ( V ) of the cone is given by, V = 1 3 π r 2 h = 1 3 π l 2 sin 2 θ l cos θ = 1 3 π l 3 sin 2 θ cos θ ∴ d V d θ = l 3 π 3 sin 2 θ ( - sin θ ) + cos θ ( 2 sin θ cos θ ) = l 3 π 3 - sin 3 + 2 sin θ cos 2 θ d 2 V d θ 2 = l 3 π 3 - 3 sin 2 θ cos θ + 2 cos 3 θ - 4 sin 2 θ cos θ = l 3 π 3 2 cos 3 θ - 7 sin 2 θ cos θ Now, d V d θ = 0 ⇒ sin 3 θ = 2 sin θ cos 2 θ ⇒ tan 2 θ = 2 ⇒ tan θ = 2 ⇒ θ = tan - 1 2 Now, when θ = tan - 1 2 , then tan 2 θ = 2 or sin 2 θ = 2 cos 2 θ Then, we have d 2 V d θ 2 = l 3 π 3 2 cos 3 θ - 14 cos 3 θ = - 4 π l 3 cos 3 θ < 0 for θ ∈ 0 , π 2 By second derivative test, the volume ( V ) is the maximum when θ = tan - 1 2 . Hence, for a given slant height, the semi-vertical angle of the cone of the maximum volume is tan - 1 2 . Hence, shown.

E M B I B E

Elements of Mathematics Class 12>Chapter 8 - Application of Derivatives>EXERCISE 8(d)>Q 9

EASY

12th Odisha Board

IMPORTANT

Earn 100

Find two numbers $x$ and $y$ whose sum is $15$ such that $x y^{2}$ is maximum.

Important Questions on Application of Derivatives

EASY

12th Odisha Board

IMPORTANT

Elements of Mathematics Class 12>Chapter 8 - Application of Derivatives>EXERCISE 8(d)>Q 10

If the sum of two positive numbers is constant then show that their product is maximum when they are equal.

EASY

12th Odisha Board

IMPORTANT

Elements of Mathematics Class 12>Chapter 8 - Application of Derivatives>EXERCISE 8(d)>Q 11

Determine a rectangle of area

25

sq. Units which has minimum perimeter.

HARD

12th Odisha Board

IMPORTANT

Elements of Mathematics Class 12>Chapter 8 - Application of Derivatives>EXERCISE 8(d)>Q 12

Find the altitude of a right circular cylinder of maximum volume inscribed in a sphere of radius

\Rightarrow x = 15

.

HARD

12th Odisha Board

IMPORTANT

Elements of Mathematics Class 12>Chapter 8 - Application of Derivatives>EXERCISE 8(d)>Q 13

Show that the radius of the right circular cylinder of greatest curved surface that can be inscribed in a given cone is half the radius of the base of the cone.

HARD

12th Odisha Board

IMPORTANT

Elements of Mathematics Class 12>Chapter 8 - Application of Derivatives>EXERCISE 8(d)>Q 14

Show that the semi vertical angle of a cone of given slant height is

\tan^{- 1} \sqrt{2}

when its volume is maximum.

EASY

12th Odisha Board

IMPORTANT

Elements of Mathematics Class 12>Chapter 8 - Application of Derivatives>EXERCISE 8(d)>Q 15

A cylindrical open water tank with a circular base is to be made out of

30

sq. metres of metal sheet. Find the dimensions so that it can hold maximum water. (Neglect thickness of sheet)

MEDIUM

12th Odisha Board

IMPORTANT

Elements of Mathematics Class 12>Chapter 8 - Application of Derivatives>EXERCISE 8(d)>Q 16

A cylindrical vessel of capacity

500

cubic metres open at the top is to be constructed. Find the dimensions of the vessel if the material used is minimum given that the thickness of the material used is