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
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`. Also find maximum volume in terms of volume of the sphere
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?
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.
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) = `x^2 + (16)/x^2`
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.
The profit function P(x) of a firm, selling x items per day is given by P(x) = (150 – x)x – 1625 . Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.
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.
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
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?
If x is real, the minimum value of x2 – 8x + 17 is ______.
The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
The combined resistance R of two resistors R1 and R2 (R1, R2 > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R1 + R2 = C (a constant), then maximum resistance R is obtained if ____________.
The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is
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 A = [aij] be a 3 × 3 matrix, where
aij = `{{:(1, "," if "i" = "j"),(-x, "," if |"i" - "j"| = 1),(2x + 1, "," "otherwise"):}`
Let a function f: R→R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to ______.
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 ______.
If the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.
If the point (1, 3) serves as the point of inflection of the curve y = ax3 + bx2 then the value of 'a ' and 'b' are ______.
The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.
The maximum distance from origin of a point on the curve x = `a sin t - b sin((at)/b)`, y = `a cos t - b cos((at)/b)`, both a, b > 0 is ______.
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.
Find the maximum and the minimum values of the function f(x) = x2ex.
Divide the number 100 into two parts so that the sum of their squares is minimum.
Find the point on the curve y2 = 4x, which is nearest to the point (2, 1).
