Advertisements
Advertisements
प्रश्न
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
Advertisements
उत्तर
Given a cylinder
S = 2πrh + πr2 .......(1)
V = πr2 h .......(2 )
From (1) S - πr2 = 2 πrh
`h = [(S -pir^2)/(2pir)] = S/(2pir) - r/2`
Put in (2)
`V = pir^2h = pir^2 [ S/(2pir) - r/2] = (Sr)/2 - (pir^3)/2`
Now diff on both sides by ‘r’
`(dv)/(dr) = S/2 - (3pir^2)/2`
For max/min `(dv)/(dr) = 0`
`S/2 - (3pir^2)/2 = 0 ⇒ S = 3pir^2` ........(3)
`(d^2 v)/(dr^2) = -3pir = -3 pi x sqrt(S/3pi) <0`
∴ By second derivative test it is maxima from (1) & (3)
`2pirh + pir^2 = 3pir^2`
`2pir h = 2 pir^2`
h = r
APPEARS IN
संबंधित प्रश्न
A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.
Find the maximum and minimum value, if any, of the following function given by g(x) = x3 + 1.
Find the maximum and minimum value, if any, of the following function given by f(x) = |sin 4x + 3|
Find the maximum and minimum value, if any, of the following function given by h(x) = x + 1, x ∈ (−1, 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) = x/2 + 2/x, x > 0`
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 two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.
Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π].
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].
Find the point on the straight line 2x+3y = 6, which is closest to the origin.
Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
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.
Choose the correct option from the given alternatives :
If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.
Solve the following : Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
Find both the maximum and minimum values respectively of 3x4 - 8x3 + 12x2 - 48x + 1 on the interval [1, 4].
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
Divide 20 into two ports, so that their product is maximum.
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is ______.
Let f(x) = |(x – 1)(x2 – 2x – 3)| + x – 3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ______.
Find the maximum and the minimum values of the function f(x) = x2ex.
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.
Sumit has bought a closed cylindrical dustbin. The radius of the dustbin is ‘r' cm and height is 'h’ cm. It has a volume of 20π cm3.
- Express ‘h’ in terms of ‘r’, using the given volume.
- Prove that the total surface area of the dustbin is `2πr^2 + (40π)/r`
- Sumit wants to paint the dustbin. The cost of painting the base and top of the dustbin is ₹ 2 per cm2 and the cost of painting the curved side is ₹ 25 per cm2. Find the total cost in terms of ‘r’, for painting the outer surface of the dustbin including the base and top.
- Calculate the minimum cost for painting the dustbin.
For a function defined on an interval \[I\], which condition means that \[f\] has a minimum value at \[c\in I\]?
Assume \[f'(c)=0\] and the second derivative exists at \[c\]. Which condition gives a local maximum?
For a continuous function on a closed interval \[[a,b]\], which values must be compared to find absolute maxima and minima?
