Advertisements
Advertisements
प्रश्न
Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base.
Advertisements
उत्तर
Let r be the radius of the circular base, h be the height and S be the total surface area of a right circular cylinder, Then S = 2πr2 + 2πrh.
Let V be the volume of the cylinder with the above dimensions.
∴ `V = pir^2h = pir^2 ((S - 2pir^2)/(2pir))`
`(∵ S = 2pir^2 + 2pirh, ∴ h = (S - 2pir^2)/(2pir))`
`= r/2 (S - 2pir^2)`
⇒ `V = (sr)/2 - pir^3`
Differentiating w.r.t. x, we get
`(dV)/(dr) = S/2- 3pir^2`
For maximum / minimum volume
`(dV)/(dr) = 0`
⇒ `S/2-3pir^2 = 0`
⇒ `r^2 = S/(6pi)`
⇒ `r = sqrt(S/(6pi))`
`(d^2V)/(dr^2) = -6pir`
and `((d^2V)/(dr^2))_(r sqrt (S/(6pi)))`
`= -6pi sqrt (S/(6pi)) < 0`
⇒ V has a maximum value at `r = sqrt (S/ (6pi))`
When `r = sqrt (S/ (6pi)), `then
`h = (S- 2pi (S/(6pi)))/ (2pi sqrt (S/ (6pi))) = (4pi (S/ (6pi)))/ (2pi sqrt (S/ (6pi)))`
⇒ `h = 2 sqrt (S/(6pi)) = 2` radius = diameter.
So volume is maximum when the height is equal to the diameter of the base.
APPEARS IN
संबंधित प्रश्न
Find the maximum and minimum value, if any, of the following function given by f(x) = −(x − 1)2 + 10
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:
f(x) = sinx − cos x, 0 < x < 2π
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) = x/2 + 2/x, x > 0`
Prove that the following function do not have maxima or minima:
g(x) = logx
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) = 4x - 1/x x^2, x in [-2 ,9/2]`
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = (x −1)2 + 3, x ∈[−3, 1]
What is the maximum value of the function sin x + cos x?
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Find two numbers whose sum is 24 and whose product is as large as possible.
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
Find the maximum area of an isosceles triangle inscribed in the ellipse `x^2/ a^2 + y^2/b^2 = 1` with its vertex at one end of the major axis.
Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π].
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?
Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.
A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get the maximum area. Also, find the maximum area.
Find the maximum and minimum of the following functions : f(x) = x log x
A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
Determine the maximum and minimum value of the following function.
f(x) = x log x
Divide the number 20 into two parts such that their product is maximum.
If f(x) = x.log.x then its maximum value is ______.
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals 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 lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is `pi/3`
The maximum value of `(1/x)^x` is ______.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is
Let x and y be real numbers satisfying the equation x2 – 4x + y2 + 3 = 0. If the maximum and minimum values of x2 + y2 are a and b respectively. Then the numerical value of a – b is ______.
The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?
The absolute maximum value of the function f(x) = 2x3 − 3x2 − 36x + 9 defined on [−3, 3] is ______.
