Advertisements
Advertisements
Question
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
Advertisements
Solution
Let the length and breadth of a rectangle be x and y, respectively.
∴ Perimeter of rectangle = 2 (x + y) = 36 cm
∴ x + y = 18
∴ y = 18 − x ....(i)
Area of rectangle = x × y
= x (18 − x) ...from eq (i)
Let f(x) = x(18 − x)
= 18x − x2
Then f'(x) = `d/dx (18 - x^2)`
= 18 × 1 − 2x
= 18 − 2x
and f''(x) `d/dx (18 - 2x)`
= 0 − 2 × 1
= −2
Now, f'(x) = 0, if 18 − 2x = 0
i.e., if x = 9
and f''(9) = −2 < 0
∴ by the second derivative test, f has a maximum value at x = 9.
When x = 9, y = 18 − 9 = 9
Hence, the rectangle is a square of side 9 cm.
RELATED QUESTIONS
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) = x + 1, x ∈ (−1, 1)
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
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 whose sum is 16 and the sum of whose cubes is minimum.
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.
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
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 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
Solve the following : Show that of all rectangles inscribed in a given circle, the square has the 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 : 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.
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.
Solve the following : 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)`.
The function f(x) = x log x is minimum at x = ______.
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
The maximum value of `(1/x)^x` is ______.
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
The maximum value of the function f(x) = `logx/x` is ______.
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 ______.
The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30° is ______.
A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?
When is a point in the domain of a function called a critical point?
Let \[c\] be a critical point of a continuous function \[f\]. If \[f'(x)\] changes sign from negative to positive as \[x\] passes through \[c\], what is \[c\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=-2\]?
