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
An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.
Find the maximum and minimum value, if any, of the following function given by g(x) = x3 + 1.
Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
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).`
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.
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.
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
The minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
The function y = 1 + sin x is maximum, when x = ______
The maximum value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
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`
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 ____________.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
A ball is thrown upward at a speed of 28 meter per second. What is the speed of ball one second before reaching maximum height? (Given that g= 10 meter per second2)
Range of projectile will be maximum when angle of projectile is
Read the following passage and answer the questions given below.
|
|
- Is the function differentiable in the interval (0, 12)? Justify your answer.
- If 6 is the critical point of the function, then find the value of the constant m.
- Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.
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.
The minimum value of 2sinx + 2cosx is ______.
Read the following passage:
Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore.
|
Based on the above information, answer the following questions:
- If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
- Find `(dV)/(dr)`. (1)
- (a) Find the radius of cylinder when its volume is maximum. (2)
OR
(b) For maximum volume, h > r. State true or false and justify. (2)
Check whether the function f : R `rightarrow` R defined by f(x) = x3 + x, has any critical point/s or not ? If yes, then find the point/s.
