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
A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.
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) = |sin 4x + 3|
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
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?
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 \[\pi : (\pi + 2)\].
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.
Solve the following : Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is `(2"R")/sqrt(3)`. Also, find the maximum volume.
Determine the maximum and minimum value of the following function.
f(x) = x log x
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.
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 ______
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`
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
Read the following passage and answer the questions given below.
|
In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of `x^2/a^2 + y^2/b^2` = 1. |
- If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
- Find the critical point of the function.
- Use First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
OR
Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
Let f: R → R be a function defined by f(x) = (x – 3)n1(x – 5)n2, n1, n2 ∈ N. Then, which of the following is NOT true?
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 ______.
The range of a ∈ R for which the function f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)cot(x/2)sin^2(x/2), x ≠ 2nπ, n∈N` has critical points, is ______.
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 minimum value of 2sinx + 2cosx is ______.
The minimum value of the function f(x) = xlogx is ______.
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.
20 is divided into two parts so that the product of the cube of one part and the square of the other part is maximum, then these two parts are
