Advertisements
Advertisements
Question
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
Advertisements
Solution
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
RELATED QUESTIONS
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.` Also, find the maximum volume.
If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.
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 the maximum and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.
Find the maximum and minimum value, if any, of the following function given by h(x) = x + 1, x ∈ (−1, 1)
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) = 1/(x^2 + 2)`
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
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]`
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.
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
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.
Find the maximum and minimum of the following functions : y = 5x3 + 2x2 – 3x.
Divide the number 20 into two parts such that sum of their squares is minimum.
A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
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 among rectangles of given area, the square has least perimeter.
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 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
A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?
The minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
The two parts of 120 for which the sum of double of first and square of second part is minimum, are ______.
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
Find all the points of local maxima and local minima of the function f(x) = `- 3/4 x^4 - 8x^3 - 45/2 x^2 + 105`
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?
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
The maximum value of `(1/x)^x` is ______.
The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
Find the points of local maxima and local minima respectively for the function f(x) = sin 2x - x, where `-pi/2 le "x" le pi/2`
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
The combined resistance R of two resistors R1 and R2 (R1, R2 > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R1 + R2 = C (a constant), then maximum resistance R is obtained if ____________.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
Divide 20 into two ports, so that their product is maximum.
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.
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is ______.
If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.
If S1 and S2 are respectively the sets of local minimum and local maximum points of the function. f(x) = 9x4 + 12x3 – 36x2 + 25, x ∈ R, then ______.
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 rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, 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)
Find the maximum and the minimum values of the function f(x) = x2ex.
A right circular cylinder is to be made so that the sum of the radius and height is 6 metres. Find the maximum volume of the cylinder.
