Advertisements
Advertisements
Question
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/cm2 and the material for the sides costs Rs 2.50/cm2. Find the least cost of the box.
Advertisements
Solution
Let x be the side of the square base and y be the length of the vertical sides.
Area of the base and bottom = 2x2 cm2
∴ Cost of the material required = ₹ 5 × 2x2
= ₹ 10x2
Area of the 4 sides = 4xy cm2
∴ Cost of the material for the four sides
= ₹ 2.50 x 4xy
= ₹ 10xy
Total cost C = 10x2 + 10xy .....(i)
New volume of the box = x × x × y
⇒ 1024 = x2y
∴ y = `1024/x^2` ....(ii)
Putting the value of y in equation (i) we get
C = `10x^2 + 10x xx 1024/x^2`
⇒ C = `10x^2 + 10240/x`
Differentiating both sides w.r.t. x, we get
`"dC"/"dx" = 20x - 10240/x^2` ....(iii)
For local maxima and local minima `"dC"/"dx"` = 0
`20 - 102400/x^2` = 0
⇒ 20x3 – 10240 = 0
⇒ x3 = 512
⇒ x = 8 cm
Now from equation (ii)
y = `10240/(8)^2`
= `10240/64`
= 16 cm
∴ Cost of material used C = 10x2 + 10xy
= 10 × 8 × 8 + 10 × 8 × 16
= 640 + 1280
= 1920
Now differentiating equation (iii) we get
`("d"^2"C")/("dx"^2) = 20 + 20480/x^3`
Put x = 8
= `20 + 20480/(8)^3`
= `20 + 20480/512`
= 20 + 40 = 60 > 0 minima
Hence, the required cost is ₹ 1920 which is the minimum.
APPEARS IN
RELATED QUESTIONS
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x).
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
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.
Find two numbers whose sum is 24 and whose product is as large as possible.
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
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 point on the straight line 2x+3y = 6, which is closest to the origin.
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.
Find the maximum and minimum of the following functions : y = 5x3 + 2x2 – 3x.
Find the maximum and minimum of the following functions : f(x) = x log x
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 ______.
Determine the maximum and minimum value of the following function.
f(x) = 2x3 – 21x2 + 36x – 20
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
By completing the following activity, examine the function f(x) = x3 – 9x2 + 24x for maxima and minima
Solution: f(x) = x3 – 9x2 + 24x
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme values, f'(x) = 0, we get
x = `square` or `square`
∴ f''`(square)` = – 6 < 0
∴ f(x) is maximum at x = 2.
∴ Maximum value = `square`
∴ f''`(square)` = 6 > 0
∴ f(x) is maximum at x = 4.
∴ Minimum value = `square`
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 f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.
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`
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.
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.
The maximum value of sin x . cos x is ______.
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
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 ______.
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 ______.
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 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)`
The minimum value of 2sinx + 2cosx is ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
If x + y = 8, then the maximum value of x2y is ______.
What is an absolute minimum?
For \[f(x)=3x^4+4x^3-12x^2+12\], what is \[f'(x)\]?
