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 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 f(x) = (2x − 1)2 + 3.
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 function given by f(x) = |x + 2| − 1.
Find the maximum and minimum value, if any, of the following function given by h(x) = x + 1, x ∈ (−1, 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:
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:
f(x) = ex
Prove that the following function do not have maxima or minima:
g(x) = logx
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
Show that the right circular cone of least curved surface and given volume has an altitude equal to `sqrt2` time the radius of the base.
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has
- local maxima
- local minima
- point of inflexion
Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π].
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
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?
Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .
Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20
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.
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 perimeter of a triangle is 10 cm. If one of the side is 4 cm. What are the other two sides of the triangle for its maximum area?
Solve the following : Show that a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.
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.
The function f(x) = x log x is minimum at x = ______.
If z = ax + by; a, b > 0 subject to x ≤ 2, y ≤ 2, x + y ≥ 3, x ≥ 0, y ≥ 0 has minimum value at (2, 1) only, then ______.
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
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 ______.
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.
If x is real, the minimum value of x2 – 8x + 17 is ______.
The maximum value of `(1/x)^x` is ______.
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
A function f(x) is maximum at x = a when f'(a) > 0.
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 ______.
Let f(x) = |(x – 1)(x2 – 2x – 3)| + x – 3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ______.
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 volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30° 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 ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
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.
If x + y = 8, then the maximum value of x2y is ______.
Divide the number 100 into two parts so that the sum of their squares is minimum.
