Advertisements
Advertisements
प्रश्न
Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.
Advertisements
उत्तर
Let ABCD be a rectangle inscribed in a circle of radius r. Let AB = x and BC = y.
Then x2 + y2 = 4r2 … (1)
Area of rectangle = xy
= `xsqrt(4r^2 - x^2)` ...[By (1)]
Let f(x) = x2(4t2 – x2)
= 4r2x2 – x4
∴ f'(x) = `d/dx(4r^2x^2 - x^4)`
= 4r2 x 2x – 4x3
= 8r2x – 4x3
and
f"(x) = `d/dx(8r^2x - 4x^3)`
= 8r2 x 1 – 4 x 3x2
= 8r2 – 12x2
For maximum area, f'(x) = 0
∴ 8r2x – 4x3 = 0
∴ 4x3 = 8r2x
∴ x2 = 2r2 ...[∵ x ≠ 0]
∴ x = `sqrt(2)r` ...[∵ x > 0]
and
`f"(sqrt(2r)) = 8r^2 – 12(sqrt(2r))`
= – 16r2 < 0
∴ f(x) is maximum when x = `sqrt(2)r`
If x = `sqrt(2)r`, then from (1),
`(sqrt(2r))^2 + y^2` = 4r2
∴ y2 = 4r2 – 2r2 = 2r2
∴ y = `sqrt(2)r` ...[∵ y > 0]
∴ x = y
∴ rectangle is a square.
Hence, amongst all rectangles inscribed in a circle, the square has maximum area.
APPEARS IN
संबंधित प्रश्न
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
Find the maximum and minimum value, if any, of the following function given by g(x) = x3 + 1.
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:
f(x) = x2
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
What is the maximum value of the function sin x + cos x?
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.
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is `8/27` of the volume of the sphere.
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 maximum and minimum of the following functions : y = 5x3 + 2x2 – 3x.
Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`
Find the maximum and minimum of the following functions : f(x) = x log x
Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.
If f(x) = x.log.x then its maximum value is ______.
Twenty meters of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is ______
The function y = 1 + sin x is maximum, when x = ______
The two parts of 120 for which the sum of double of first and square of second part is minimum, are ______.
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is `pi/3`
An open box with square base is to be made of a given quantity of cardboard of area c2. Show that the maximum volume of the box is `"c"^3/(6sqrt(3))` cubic units
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.
If x is real, the minimum value of x2 – 8x + 17 is ______.
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
The maximum value of sin x . cos x is ______.
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`
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
The function `"f"("x") = "x" + 4/"x"` has ____________.
The function `f(x) = x^3 - 6x^2 + 9x + 25` has
The maximum value of the function f(x) = `logx/x` is ______.
Divide 20 into two ports, so that their product is maximum.
Let x and y be real numbers satisfying the equation x2 – 4x + y2 + 3 = 0. If the maximum and minimum values of x2 + y2 are a and b respectively. Then the numerical value of a – b is ______.
The set of values of p for which the points of extremum of the function f(x) = x3 – 3px2 + 3(p2 – 1)x + 1 lie in the interval (–2, 4), is ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
The rectangle has area of 50 cm2. Complete the following activity to find its dimensions for least perimeter.
Solution: Let x cm and y cm be the length and breadth of a rectangle.
Then its area is xy = 50
∴ `y =50/x`
Perimeter of rectangle `=2(x+y)=2(x+50/x)`
Let f(x) `=2(x+50/x)`
Then f'(x) = `square` and f''(x) = `square`
Now,f'(x) = 0, if x = `square`
But x is not negative.
∴ `x = root(5)(2) "and" f^('')(root(5)(2))=square>0`
∴ by the second derivative test f is minimum at x = `root(5)(2)`
When x = `root(5)(2),y=50/root(5)(2)=root(5)(2)`
∴ `x=root(5)(2) "cm" , y = root(5)(2) "cm"`
Hence, rectangle is a square of side `root(5)(2) "cm"`
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.
A right circular cylinder is to be made so that the sum of the radius and height is 6 metres. Find the maximum volume of the cylinder.
Divide the number 100 into two parts so that the sum of their squares is minimum.
A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?
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`
Which statement gives the meaning of a local maximum at \[c\]?
Which statement gives the meaning of a local minimum at \[c\]?
What is an absolute maximum?
Let \[c\] be a critical point of a continuous function \[f\]. If \[f'(x)\] changes sign from negative to positive as \[x\] passes through \[c\], what is \[c\]?
For a continuous function on a closed interval \[[a,b]\], which values must be compared to find absolute maxima and minima?
