Advertisements
Advertisements
प्रश्न
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
उत्तर
Let the length and breadth of a rectangle be x cm and y cm
∴ Perimeter of rectangle = 2(x + y) = 120 cm
∴ x + y = 60 .......(i)
Let A be the area of the rectangle.
∴ A = xy
= x(60 − x) .......[From (i)]
= 60x − x2
Differentiating w. r. t. x, we get
`("dA")/("d"x)` = 60 − 2x
∴ `("d"^2"A")/("d"x^2)` = −2
For maximum area, `"dA"/("d"x)` = 0
∴ 60 − 2x = 0
∴ x = 30
For x = 30,
`(("d"^2"A")/("d"x^2))_(x = 30)` = − 2 < 0
When x = 30, area of the rectangle is maximum.
and y = 60 − 30 = 30 .......[From (i)]
∴ Area of the rectangle is maximum if length = breadth = 30 cm.
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 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:
`h(x) = sinx + cosx, 0 < x < pi/2`
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) = x/2 + 2/x, x > 0`
Prove that the following function do not have maxima or minima:
g(x) = logx
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) =x^3, x in [-2,2]`
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]`
What is the maximum value of the function sin x + cos x?
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 the maximum and minimum values of x + sin 2x on [0, 2π].
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
Find the maximum area of an isosceles triangle inscribed in the ellipse `x^2/ a^2 + y^2/b^2 = 1` with its vertex at one end of the major axis.
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).`
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.`
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.
Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .
Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.
A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle.
Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
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.
A box with a square base is to have an open top. The surface area of the box is 192 sq cm. What should be its dimensions in order that the volume is largest?
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 the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
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:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
Determine the maximum and minimum value of the following function.
f(x) = 2x3 – 21x2 + 36x – 20
Determine the maximum and minimum value of the following function.
f(x) = x log x
A metal wire of 36cm long is bent to form a rectangle. Find it's dimensions when it's area is maximum.
If f(x) = x.log.x then its maximum value is ______.
If x + y = 3 show that the maximum value of x2y is 4.
Divide the number 20 into two parts such that their product 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`
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
The function y = 1 + sin x is maximum, when x = ______
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
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.
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?
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?
The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
Find the area of the largest isosceles triangle having a perimeter of 18 meters.
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^3 - 6x^2 + 9x + 25` has
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is
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 ______.
A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then `(4/π + 1)`k is equal to ______.
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 ______.
If the point (1, 3) serves as the point of inflection of the curve y = ax3 + bx2 then the value of 'a ' and 'b' are ______.
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 ______.
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
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 ______.
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 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 minimum value of 2sinx + 2cosx is ______.
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 ______.
A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane bounded by the lines y = 0, y = 3x and y = 30 – 2x. The largest area of such a rectangle is ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.
Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x2, where x is the number of units and P is the profit in rupees.
