Advertisements
Advertisements
Question
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?
Advertisements
Solution
The base of the tank is square.
Let the length, width and height of the open tank be x, x and y units respectively.
Volume = Length × Breadth × Height = x2 y
Total surface area = 2(lb + bh + hl) − lb = x2 + 4xy.
The volume of the tank is given to be constant
Now, surface area = x2 + 4xy
For the total surface area to be least
Hence, the surface area is minimum when x = 2y, i.e., the depth of the tank is half of its width.
Now if the surface area of the sheet is minimum the cost of the sheet will be least as well, Thus making the tank economical and cost-effective.
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).
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 maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 2
Find the maximum and minimum value, if any, of the function given by f(x) = |x + 2| − 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) = x3 − 3x
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
Find the maximum and minimum values of x + sin 2x on [0, 2π].
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?
Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base.
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?
Show that the right circular cone of least curved surface and given volume has an altitude equal to `sqrt2` time the radius of the base.
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`
The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is Rs. 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it.
Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`
Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.
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.
Solve the following : 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)`.
If f(x) = x.log.x then its maximum value is ______.
If x + y = 3 show that the maximum value of x2y is 4.
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.
Find both the maximum and minimum values respectively of 3x4 - 8x3 + 12x2 - 48x + 1 on the interval [1, 4].
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 function `"f"("x") = "x" + 4/"x"` has ____________.
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 ____________.
Range of projectile will be maximum when angle of projectile is
The function `f(x) = x^3 - 6x^2 + 9x + 25` has
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
The minimum value of the function f(x) = xlogx is ______.
If Mr. Rane order x chairs at the price p = (2x2 - 12x - 192) per chair. How many chairs should he order so that the cost of deal is minimum?
Solution: Let Mr. Rane order x chairs.
Then the total price of x chairs = p·x = (2x2 - 12x- 192)x
= 2x3 - 12x2 - 192x
Let f(x) = 2x3 - 12x2 - 192x
∴ f'(x) = `square` and f''(x) = `square`
f'(x ) = 0 gives x = `square` and f''(8) = `square` > 0
∴ f is minimum when x = 8
Hence, Mr. Rane should order 8 chairs for minimum cost of deal.
