Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let x be the radius of the base, h be the height, V be the volume and S be the total surface area of the cylindrical tank.
Then V = `pia^3` ...(Given)
∴ `pix^2h = pia^3`
∴ h = `a^3/x^2` ...(1)
Now, S = `2pixh + pix^2`
= `2pix(a^3/x^2) + pix^2` ...[By (1)]
= `(2pia^3)/x + pix^2`
∴ `"dS"/dx = d/dx((2pia^3)/ x + pix^2)`
= 2πa3 (– 1)x–2 + π x 2x
= `(-2pia^3)/x^2 + 2pix`
and
`(d^2S)/(dx^2) = d/dx((-2pia^3)/x^2 + 2pix)`
= – 2πa3(– 2)x–3 + 2π x 1
= `(4pia^3)/x^3 + 2pi`
Now, `"dS"/dx = 0 "gives" (-2pia^3)/x^2 + 2pix` = 0
∴ – 2πa3 + 2πx3 = 0
∴ 2πx3 = 2πa3
∴ x = a
and
`((d^2S)/dx^2)_("at" x = a)`
= `(4pia^3)/a^3 + 2pi`
= 6π > 0
∴ by the second derivative test,
S is minimum when x = a
When x = a, from (1)
h = `a^3/a^2` = a
Hence, the quantity of metal sheets is minimum when radius = height = a cm.
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 maximum and minimum value, if any, of the function given by f(x) = |x + 2| − 1.
Find the maximum and minimum value, if any, of the following function given by f(x) = |sin 4x + 3|
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) = x3 − 3x
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) = x3 − 6x2 + 9x + 15
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)`
It is given that at x = 1, the function x4− 62x2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.
A square piece of tin of side 18 cm is to made into a box without a top by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
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?
A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.
Show that the minimum length of the hypotenuse is `(a^(2/3) + b^(2/3))^(3/2).`
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 among rectangles of given area, the square has least perimeter.
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 : Show that of all rectangles inscribed in a given circle, the square has the maximum area.
Solve the following : Show that a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.
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:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
Determine the maximum and minimum value of the following function.
f(x) = x log x
If x + y = 3 show that the maximum value of x2y is 4.
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
Divide the number 20 into two parts such that their product is maximum
If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.
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 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.
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`
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
Find the points of local maxima, local minima and the points of inflection of the function f(x) = x5 – 5x4 + 5x3 – 1. Also find the corresponding local maximum and local minimum values.
The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and `x/3` and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.
The maximum value of `(1/x)^x` is ______.
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
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)
Divide 20 into two ports, so that their product is maximum.
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?
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 sum of all the local minimum values of the twice differentiable function f : R `rightarrow` R defined by
f(x) = `x^3 - 3x^2 - (3f^('')(2))/2 x + f^('')(1)`
Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.
Divide the number 100 into two parts so that the sum of their squares is minimum.
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?
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) `= x sqrt(1 - x), 0 < x < 1`
