Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
Let the length, breadth and height of the metal box be x cm, x cm and y cm respectively.
It is given that the box can contain 1024 cm3.
∴ 1024 = x2y
`=> y = 1024/x^2` .....(1)
Let C be the cost in rupees of the material used to construct.
Then
`C = 5x^2+5x^2 + 5/2 xx 4xy`
`C = 10x^2 + 10xy`
We have to find the least value of C.
`C = 10x^2 + 10xy`
`C = 10x^2 + 10x xx 1024/x^2`
`C = 10x^2 + 10240/x`
`=> (dC)/(dx) = 20x - 10240/x^2`
And
`=> (d^2C)/(dx^2) = 20 + 20480/x^3`
The Critical number for C are given by `(dC)/(dx) = 0`
Now
`=> (dC)/(dx) = 0`
`=> 20x - 10240/x^2 = 0`
`=> x^3 = 512`
`=> x = 8`
Also `((d^2C)/(dx^2))_(x = 8) = 20 + 20480/8^3 >0`
Thus, the cost of the box is least when x = 8.
Put x = 8 in (1), we get y = 16.
So, dimensions of the box are 8 × 8 × 16
Put x = 8, y = 16 in C = 10x2 + 10xy, we get C = 1920
Hence the least cost of the box is 1920
APPEARS IN
संबंधित प्रश्न
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x).
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 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 absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) =x^3, x in [-2,2]`
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
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?
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 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.
Divide the number 20 into two parts such that sum of their squares is minimum.
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?
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also, find the maximum volume.
Find both the maximum and minimum values respectively of 3x4 - 8x3 + 12x2 - 48x + 1 on the interval [1, 4].
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
Range of projectile will be maximum when angle of projectile 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.
If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
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 ______.
The lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle a. with the plane of the base. The value of a for which the volume of the pyramid is greatest, is ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` 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 ______.
Find the maximum and the minimum values of the function f(x) = x2ex.
20 is divided into two parts so that the product of the cube of one part and the square of the other part is maximum, then these two parts are
