Advertisements
Advertisements
Question
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.
Advertisements
Solution
Let ABCD be the rectangle inscribed in a semicircle of radius 1 unit such that the vertices A and B lie on the diameter.
Let AB = DC = x and BC = AD = y.
Let O be the centre of the semicircle.
Join OC and OD. Then OC = OD = radius = 1.
Also, AD = BC and m∠A = m∠B = 90°.
∴ OA= OB
∴ OB = `(1)/(2) "AB" = x/(2)`
In right angled triangle OBC,
OB2 + BC2 = OC2
∴ `(x/2)^2 + y^2` = 12
∴ y2 = `1 - x^2/(4) = (1)/(4)(4 - x^2)`
∴ y = `(1)/(2)sqrt(4 - x^2)` ...[∵ y > 0]
Also of the triangle
= xy
= `x.(1)/(2)sqrt(4 - x^2)`
Let f(x) = `(1)/(2) xx sqrt(4 - x^2)`
= `(1)/(2)sqrt(4x2 - x^4)`
∴ f'(x) = `(1)/(2)d/dx(sqrt(4x^2 - x^4))`
= `(1)/(2) xx (1)/(2sqrt(4x^2 - x^4)) xx.d/dx(4x^2 - x^4)`
= `(1)/(4sqrt(4x^2 - x^4)) xx (4 xx 2x - 4x^3)`
= `(4x(2 - x2))/(4xsqrt(4 - x^2)`
= `(2 - x^2)/sqrt(4 - x^2)` ...[∵ x ≠ 0]
and
f"(x) = `d/dx((2 - x^2)/sqrt(4 - x^2))`
= `d/dx[((4 - x^2) - 2)/sqrt(4 - x^2)]`
= `d/dx[sqrt(4 - x^2) - (2)/sqrt(4 - x^2)]`
= `d/dx(sqrt(4 - x^2)) - 2d/dx(4 - x^2)^(-1/2)`
= `(1)/(2sqrt(4 - x^2)).d/dx(4- x^2) - 2(-1/2)(4 - x^2)^(-3/2).d/dx(4 - x^2)`
= `(1)/(2sqrt(4 -+ x^2)) xx (0 - 2x) + (1)/(4 - x^2)^(3/2) xx (0 - 2x)`
= `-x/sqrt(4 - x^2) - (2x)/(4 - x^2)^(3/2)`
= `(-x(4 - x^2) - 2x)/(4 - x^2)^(3/2)`
= `(-4x + x^3 - 2x)/(4 - x^2)^(3/2)`
= `(x^3 - 6x)/(4 - x^2)^(3/2)`
For maximum value of f(x),f'(x) = 0
∴ `(2 - x^2)/sqrt(4 - x^2)` = 0
∴ 2 – x2 = 0
∴ x2 = 2
∴ x = `sqrt(2)` ...[∵ x > 0]
Now, f"`(sqrt(2)) = ((sqrt(2))^3 - 6sqrt(2))/[4 - (sqrt(2))^2]^(3/2)`
= `(-4sqrt(2))/(2sqrt(2))`
= – 2 < 0
∴ by the second derivative test, f is maximum when x = `sqrt(2)`
When `x = sqrt(2), y = (1)/(2)sqrt(4 - x^2)`
= `(1)/(2)sqrt(4 - 2)`
= `(1)/(2) xx sqrt(2)`
= `(1)/sqrt(2)`
∴ `x = sqrt(2) and y = (1)/sqrt(2)`
Hence, the area of the rectangle is maximum (i.e. rectangle has the largest size) when its length is `sqrt(2)` units and breadth is `(1)/sqrt(2)`unit.
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).
Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.` Also, find the maximum volume.
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
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
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) = 1/(x^2 + 2)`
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) = 4x - 1/x x^2, x in [-2 ,9/2]`
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.
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.
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).`
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?
A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get the maximum area. Also, find the maximum area.
Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x
Find the maximum and minimum of the following functions : f(x) = x log x
Divide the number 20 into two parts such that sum of their squares is minimum.
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.
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.
Show that among rectangles of given area, the square has least perimeter.
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.
Determine the maximum and minimum value of the following function.
f(x) = x log x
State whether the following statement is True or False:
An absolute maximum must occur at a critical point or at an end point.
If x + y = 3 show that the maximum value of x2y is 4.
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
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`
If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
The function y = 1 + sin x is maximum, when x = ______
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`
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
Find the points of local maxima, local minima and the points of inflection of the function f(x) = x5 – 5x4 + 5x3 – 1. Also find the corresponding local maximum and local minimum values.
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.
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.
The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and `x/3` and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
Find the area of the largest isosceles triangle having a perimeter of 18 meters.
If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
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 lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle a. with the plane of the base. The value of a for which the volume of the pyramid is greatest, is ______.
The minimum value of 2sinx + 2cosx is ______.
A straight line is drawn through the point P(3, 4) meeting the positive direction of coordinate axes at the points A and B. If O is the origin, then minimum area of ΔOAB is equal to ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
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.
