Advertisements
Advertisements
प्रश्न
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?
उत्तर
Let the length of one piece be x m and other piece is of length (28 - x) m Let the length of the piece bent into the shape of a circle be x m and length of the other piece bent into the shape of a square is (28 - x) m.
Circumference = 2πr
⇒ 2πr = x
⇒ `r = x/(2pi)`
Area of the circle= π (radius)2
`= pi (x/(2pi))^2 = x^2/(4pi)`
Perimeter of square = 4 side
⇒ 28 - x = 4 side
⇒ side = `(28 - x)/4`
⇒ Area of the square = (side)2
`= ((28 - x)/4)^2`
`= (28 - x)^2/16`
Let A be the sum of the areas of the two figures, then
`A = x^2/(4pi) + (28 - x)^2/16`
Differentiating w.r.t. x, we get
`(dA)/dx = (2x)/(4pi) + (2 (28 - x)(-1))/16`
`= x/(2pi) - (28 - x)/8`
For maximum / minimum, `(dA)/dx = 0`
⇒ `x / (2pi) - (28 - x)/8 = 0`
⇒ ` (4x - 28pi + xpi)/(8pi) = 0`
⇒ `4x + xpi = 28 pi`
⇒ `x = (28pi)/ (4 + pi)`
⇒ `(d^2A)/dx^2 = 1/(2pi) - (-1)/8 = 1/ (2pi) + 1/8`
and `((d^2A)/dx^2)_(x = (28pi)/(4+pi))`
`= 1/(2pi) + 1/8 > 0`
Hence area A is minimum
∴ The wire must be cut at a distance of `(28pi)/(4+pi)` m. from one end.
Hence, the length of the two pieces are `(28pi)/(4 + pi)` m and `(28 - (28pi)/(4+pi)) m 112/(4 + pi)` m
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 f(x) = (2x − 1)2 + 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 maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
What is the maximum value of the function sin x + cos x?
Find two numbers whose sum is 24 and whose product is as large as possible.
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
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.
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.
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?
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.
Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.
Solve the following : A window is in the form of a rectangle surmounted by a semicircle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.
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)`.
Solve the following : Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is `(2"R")/sqrt(3)`. Also, find the maximum volume.
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
Determine the maximum and minimum value of the following function.
f(x) = x log x
The function f(x) = x log x is minimum at x = ______.
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
Divide the number 20 into two parts such that their product is maximum
The minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.
The function y = 1 + sin x is maximum, when x = ______
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
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 the points of local maxima and local minima respectively for the function f(x) = sin 2x - x, where `-pi/2 le "x" le pi/2`
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` 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 ______.
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.
A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, is ______.
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
A running track of 440 m is to be laid out enclosing a football field. The football field is in the shape of a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum,then find the length of its sides. Also calculate the area of the football field.
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) `= x sqrt(1 - x), 0 < x < 1`
