Advertisements
Advertisements
Question
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
Advertisements
Solution
Let the length and breadth of a rectangle be l and b respectively.
∴ Perimeter of rectangle = 2(l + b) = 108 m
∴ l + b = 54
∴ b = 54 – l ...(i)
Area of rectangle = l × b
= l(54 – l) ...[From (i)]
Let f(l) = l(54 – l)
= 54l – l2
∴ f'(l) = 54 – 2l
∴ f"(l) = – 2
Consider, f'(l) = 0
∴ 54 – 2l = 0
∴ 54 = 2l
∴ l = 27
For l = 27,
f"(27) = – 2 < 0
∴ f(l) i.e., area is maximum at l = 27
and b = 54 – 27 ...[From (i)]
= 27
∴ The dimensions of rectangle are 27 m × 27 m.
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.
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`. Also find maximum volume in terms of volume of the sphere
Find two numbers whose sum is 24 and whose product is as large as possible.
Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.
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
Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .
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) = `x^2 + (16)/x^2`
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:
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.
Divide the number 20 into two parts such that their product is maximum
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
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 ______
An open box with square base is to be made of a given quantity of cardboard of area c2. Show that the maximum volume of the box is `"c"^3/(6sqrt(3))` cubic units
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.
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
Find both the maximum and minimum values respectively of 3x4 - 8x3 + 12x2 - 48x + 1 on the interval [1, 4].
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
The minimum value of the function f(x) = xlogx 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 and the minimum values of the function f(x) = x2ex.
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.
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) `= x sqrt(1 - x), 0 < x < 1`
