Advertisements
Advertisements
प्रश्न
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
Advertisements
उत्तर
Let x be the radius of base, h be the height and S be the surface area of the closed right circular cylinder whose volume is V which is given to be constant.
Then `pir^2h` = V
∴ h = `"V"/(pir^2) = "A"/x^2`, ...(1)
where A = `"V"/(pir^2)`, which is constant.
Now, S = 2πxh + 2πx2
= `2pix ("A"/x^2) + 2pix^2` ...[By (1)]
= `(2pi"A")/x + 2pix^2`
∴ `"dS"/dx = d/dx((2pi"A")/x + 2pix^2)`
= `2pi"A"(- 1)x^-2 + 2pi xx 2x`
= `(-2pi"A")/x^2 + 4pix`
and
`(d^2S)/(dx^2) = d/dx((-2pi"A")/x^2 + 4pix)`
= `-2pi"A"(-2)x^-3 + 4pi xx 1`
= `(4pi"A")/x^3 + 4pi`
Now, `"dS"/dx = 0 "gives" (-2pi"A")/x^2 + 4pix`= 0
∴ `-2pi + 4pix^3` = 0
∴ `4pixx^3 = 2pi"A"`
∴ x3 = `"A"/(2)`
∴ x = `("A"/2)^(1/3)`
and
`((d^2S)/(dx^2))_("at" x = ("A"/2)^(1/3)`
= `(4pi"A")/(("A"/2)) + 4pi`
= 12π > 0
∴ by the second derivative test, S is minimum when x = `("A"/2)^(1/3)`
When x = `("A"/2)^(1/3)`, from (1),
h = `"A"/(("A"/2)^(2/3)`
= `2^(2/3)."A"^(1/3)`
= `2.("A"/2)^(1/3)`
∴ h = 2x
Hence, the surface area is least when height of the closed right circular cylinder is equal to its diameter.
APPEARS IN
संबंधित प्रश्न
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
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 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:
`h(x) = sinx + cosx, 0 < x < pi/2`
Prove that the following function do not have maxima or minima:
f(x) = ex
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = sin x + cos x , x ∈ [0, π]
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
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 metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs 5 per cm2 and the material for the sides costs Rs 2.50 per cm2. Find the least cost of the box
Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .
Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.
Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x
Divide the number 30 into two parts such that their product is maximum.
Show that among rectangles of given area, the square has least perimeter.
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 altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/(3)`.
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.
The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
If z = ax + by; a, b > 0 subject to x ≤ 2, y ≤ 2, x + y ≥ 3, x ≥ 0, y ≥ 0 has minimum value at (2, 1) only, then ______.
The maximum value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
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?
AB is a diameter of a circle and C is any point on the circle. Show that the area of ∆ABC is maximum, when it is isosceles.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
Find the points of local maxima and local minima respectively for the function f(x) = sin 2x - x, where `-pi/2 le "x" le pi/2`
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
Range of projectile will be maximum when angle of projectile is
The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is
Read the following passage and answer the questions given below.
|
|
- Is the function differentiable in the interval (0, 12)? Justify your answer.
- If 6 is the critical point of the function, then find the value of the constant m.
- Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.
Let f: R → R be a function defined by f(x) = (x – 3)n1(x – 5)n2, n1, n2 ∈ N. Then, which of the following is NOT true?
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 ______.
A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane bounded by the lines y = 0, y = 3x and y = 30 – 2x. The largest area of such a rectangle is ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.
Check whether the function f : R `rightarrow` R defined by f(x) = x3 + x, has any critical point/s or not ? If yes, then find the point/s.
Find the maximum and the minimum values of the function f(x) = x2ex.
A right circular cylinder is to be made so that the sum of the radius and height is 6 metres. Find the maximum volume of the cylinder.
Find the point on the curve y2 = 4x, which is nearest to the point (2, 1).
Mrs. Roy designs a window in her son’s study room so that the room gets maximum sunlight. She designs the window in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the window that will admit maximum sunlight into the room.
