Advertisements
Advertisements
Question
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is `8/27` of the volume of the sphere.
Advertisements
Solution
Let VAB be the volume of the largest cone contained in the sphere.
Obviously, for maximum volume, the axis of the cone should be along the height of the sphere.
Let, ∠AOC = θ
∴ AC, the radius of the base of the cone = R sin θ, where R
is the radius of the sphere.
Height of the cone VC = VO + OC = R + R cos θ
Volume of a cone; V = `1/3 pi (AC)^2 xx (VC)`
`=> V = 1/3 piR^2 sin^2 θ (R + R cos theta)`
`=> V = 1/3 piR^3 sin^2 theta (1 + cos theta)`
On differentiating
`therefore dV/(d theta) = 1/3 piR^3 [sin^2 theta (- sin theta) + (1 cos theta) * 2 sin theta cos theta]`
`= 1/3 piR^3 [- sin^3 theta + 2 sin theta cos theta + 2 sin theta (1 - sin^2 theta)]`
`= 1/3 piR^3 [- sin^3 theta + 2 sin theta cos theta + 2 sin theta - 2sin^2 theta]`
`= 1/3 pi R^3 [- 3 sin^3 theta + 2 sin theta + 2 sin theta cos theta]`
For minimum and maximum, `(dV)/(d theta) = 0`
`=> 1/3 pi"R"^3 (- 3 sin^3 theta + 2 sin cos theta + 2 sin theta)` = 0
= - 3 sin3 θ + 2 sin θ cos θ + 2 sin θ = 0
= sin θ (- 3 sin2 θ + 2 cos θ + 2) = 0
= - 3 sin2 θ + 2 cos θ +2 = 0 ...[∵ sin θ ≠ 0]
= -3 (1 - cos2 θ) + 2 cos θ + 2 = 0
= - 3 + cos2 θ + 2 cos θ + 2 = 0
⇒ 3 cos2 θ + 2 cos θ - 1 = 0
⇒ (3 cos θ - 1)(cos θ + 1) = 0
⇒ cos θ = `1/3` cos θ = - 1
But cos θ ≠ 1 because cos θ = - 1 ⇒ θ = π which is not possible.
`therefore cos theta = 1/3`
When `cos theta = 1/3`, then `sin theta = sqrt(1 - cos^2 theta) = sqrt(1 - 1/9)`
`= sqrt(8/9)`
`= (2 sqrt2)/3`
Now `(dV)/(d theta) = 1/3 piR^3 sin theta [- 3 sin^2 theta + 2 + 2 cos theta]`
`= 1/3 piR^3 sin theta (3 cos theta - 1)(cos theta + 1)`
The sign of `cos theta = 1/3, (dV)/(d theta)` changes from +ve to -ve.
∴ V is highest at `theta = cos^-1 (1/3)`.
On decreasing θ, cos θ increases.
Now cos θ = `1/3` then V is maximum.
The height of the cone for this value of cos θ is
VC = R + R cos θ
`= R + R xx 1/3 = (4R)/3`
Radius of cone = AC = R sin θ = `R * (2sqrt2)/3 = (2 sqrt2)/3 R`
∴ Maximum volume of the cone is V
`= 1/3 pi (AC)^2 (VC)`
`= 1/3 pi ((2 sqrt(2R))/3)^2 ((4R')/3)`
`= 1/3 pi xx 8/9 R^2 xx (4R)/3`
`= 8/27 (4/3 piR^3)`
`= 8/27 xx` Volume of sphere
APPEARS IN
RELATED QUESTIONS
If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.
Find the maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 2
Find the maximum and minimum value, if any, of the following function given by g(x) = x3 + 1.
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
`g(x) = 1/(x^2 + 2)`
Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π].
Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20
An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of `pia^3`cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.
A box with a square base is to have an open top. The surface area of the box is 192 sq cm. What should be its dimensions in order that the volume is largest?
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a2. Show that the maximum volume of the box is `a^3/(6sqrt(3)`.
Solve the following : A window is in the form of a rectangle surmounted by a semicircle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.
Solve the following : Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is `(2"R")/sqrt(3)`. Also, find the maximum volume.
If x + y = 3 show that the maximum value of x2y is 4.
A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
Find all the points of local maxima and local minima of the function f(x) = `- 3/4 x^4 - 8x^3 - 45/2 x^2 + 105`
If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
If y = x3 + x2 + x + 1, then y ____________.
The function `"f"("x") = "x" + 4/"x"` has ____________.
The combined resistance R of two resistors R1 and R2 (R1, R2 > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R1 + R2 = C (a constant), then maximum resistance R is obtained if ____________.
A ball is thrown upward at a speed of 28 meter per second. What is the speed of ball one second before reaching maximum height? (Given that g= 10 meter per second2)
The range of a ∈ R for which the function f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)cot(x/2)sin^2(x/2), x ≠ 2nπ, n∈N` has critical points, is ______.
Let A = [aij] be a 3 × 3 matrix, where
aij = `{{:(1, "," if "i" = "j"),(-x, "," if |"i" - "j"| = 1),(2x + 1, "," "otherwise"):}`
Let a function f: R→R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to ______.
Let P(h, k) be a point on the curve y = x2 + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P is ______.
If the point (1, 3) serves as the point of inflection of the curve y = ax3 + bx2 then the value of 'a ' and 'b' are ______.
A cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere is ______.
A straight line is drawn through the point P(3, 4) meeting the positive direction of coordinate axes at the points A and B. If O is the origin, then minimum area of ΔOAB is equal to ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
The rectangle has area of 50 cm2. Complete the following activity to find its dimensions for least perimeter.
Solution: Let x cm and y cm be the length and breadth of a rectangle.
Then its area is xy = 50
∴ `y =50/x`
Perimeter of rectangle `=2(x+y)=2(x+50/x)`
Let f(x) `=2(x+50/x)`
Then f'(x) = `square` and f''(x) = `square`
Now,f'(x) = 0, if x = `square`
But x is not negative.
∴ `x = root(5)(2) "and" f^('')(root(5)(2))=square>0`
∴ by the second derivative test f is minimum at x = `root(5)(2)`
When x = `root(5)(2),y=50/root(5)(2)=root(5)(2)`
∴ `x=root(5)(2) "cm" , y = root(5)(2) "cm"`
Hence, rectangle is a square of side `root(5)(2) "cm"`
If x + y = 8, then the maximum value of x2y is ______.
The absolute maximum value of the function f(x) = 2x3 − 3x2 − 36x + 9 defined on [−3, 3] is ______.
Which statement gives the meaning of a local maximum at \[c\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], which is the correct factorization of \[f'(x)\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=1\]?
