Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let x be the length, y be the breadth of the rectangle and r be the radius of the semicircle. Then perimeter of the window
Then perimeter of the window = x + 2y + πr, where x = 2r
This is given to be m
∴ 2r + 2y + πr = 30
2y = 30 – (π + 2)r
∴ y = `15 - ((pi + 2)r)/(2)` ...(1)
The greatest possible amount of light may be admitted if the area of the window is maximum. Let A be the area of the window.
Then A = `xy + (pir^2)/(2)`
= `2yr + (pir^2)/(2)` ...[∵ x = 2r]
= `2r[15 - ((pi + 2))/2] + (pir^2)/(2)` ...[By (1)]
= `30r - (pi + 2)r^2 + pi/(2)r^2`
= `30r - (pi + 2 - pi/2)r^2`
∴ A = `30r - ((pi + 4)/2)r^2`
∴ `"dA"/"dr" = d/"dr"[30r - ((pi + 4)/2)r^2]`
= `30 xx 1 - ((pi + 4)/2) xx 2r`
= 30 – (π + 4)r
and
`(d^2"A")/"dr" = d/"dr"[30 -(pi + 4)r]`
= `0 - (pi + 4) xx 1`
= – (π + 4)
For maximum A, `"dA"/"dr"` = 0
∴ 30 – (π + 4)r = 0
∴ r = `(30)/(pi + 4)`
and
`((d^2"A")/("dr"))_("at" r = (30)/(pi + 4)) = - (pi + 4) < 0`
∴ A is s maximum when r = `(30)/(pi + 4)`
When r = `(30)/(pi + 4) x = 2 = (60)/(pi + 4)`
and
y = `15 - ((pi + 2))/(2) xx (30)/(pi + 4)` ...[By (1)]
= `(30pi + 120 - 30pi - 60)/(2(pi + 4)`
= `(30)/(pi + 4)`
Hence, the required dimensions of the window are as follows :
Length of rectangle = `((60)/(pi + 4))` metres,
breadth of rectangle = `((30)/(pi + 4))` metres and
radius of the semicircle = `((30)/(pi + 4))` metres.
APPEARS IN
संबंधित प्रश्न
Find the maximum and minimum value, if any, of the following function given by f(x) = −(x − 1)2 + 10
Find the maximum and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.
Find the maximum and minimum value, if any, of the following function given by f(x) = |sin 4x + 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) = 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
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) = x3 − 6x2 + 9x + 15
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)`
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = sin x + cos x , x ∈ [0, π]
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]`
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`
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 per cm2 and the material for the sides costs Rs 2.50 per cm2. Find the least cost of the box
Find the maximum and minimum of the following functions : f(x) = x log x
Find the maximum and minimum of the following functions : f(x) = `logx/x`
A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
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.
A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
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) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
The function y = 1 + sin x is maximum, when x = ______
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?
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.
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 maximum value of `(1/x)^x` is ______.
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 ____________.
Range of projectile will be maximum when angle of projectile is
Let A = [aij] be a 3 × 3 matrix, where
aij = `{{:(1, "," if "i" = "j"),(-x, "," if |"i" - "j"| = 1),(2x + 1, "," "otherwise"):}`
Let a function f: R→R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to ______.
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
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 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 maximum distance from origin of a point on the curve x = `a sin t - b sin((at)/b)`, y = `a cos t - b cos((at)/b)`, both a, b > 0 is ______.
The minimum value of the function f(x) = xlogx is ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
Divide the number 100 into two parts so that the sum of their squares is minimum.
If \[\mathrm{A}+\mathrm{B}=\frac{\pi}{2}\] then the maximum value of cosA.cosB is
