Advertisements
Advertisements
प्रश्न
Mrs. Roy designs a window in her son’s study room so that the room gets maximum sunlight. She designs the window in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the window that will admit maximum sunlight into the room.
उत्तर
Let x and y be the window dimensions and x be the side of the equilateral portion.
Let A be the complete area of the window (through which light enters):
A = `xy + sqrt(3)/4 x^2`
Also, x + 2y + 2x = 12 ...(Given)
`\implies` 3x + 2y = 12
`\implies y = (12 - 3x)/2`
Then, A = `x xx ((12 - 3x)/2) + sqrt(3)/4x^2`
= `6x - (3x^2)/2 + sqrt(3)/4x^2`
Then, `(dA)/dx = 6 - 3x + sqrt(3)/2x`
For maximum light to enter, the area of the window should be the maximum
Put `(dA)/dx = 0`
`6 - 3x + sqrt(3)/2x = 0`
`x = 12/(6 - sqrt(3))`
Again, `(d^2A)/(dx^2) = -3 + sqrt(3)/2 < 0` ...(For any value of x)
i.e., A is maximum if `x = 12/(6 - sqrt(3))` and
`y = (12 - 3(12/(6 - sqrt(3))))/2`
= `(18 - 6sqrt(3))/(6 - sqrt(3))`
Hence dimensions are `(12/(6 - sqrt(3)))m`.
and `((18 - 6sqrt(3))/(6 - sqrt(3)))m`.
APPEARS IN
संबंधित प्रश्न
Find the maximum and minimum value, if any, of the function given by f(x) = |x + 2| − 1.
Prove that the following function do not have maxima or minima:
g(x) = logx
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
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?
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].
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.
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
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.
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.
Solve the following:
A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum volume. Find the lengths of the rectangular sheet of paper.
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 ______
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`
A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?
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.
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
Divide 20 into two ports, so that their product is maximum.
The minimum value of α for which the equation `4/sinx + 1/(1 - sinx)` = α has at least one solution in `(0, π/2)` is ______.
If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
The minimum value of 2sinx + 2cosx 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 ______.
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
