Advertisements
Advertisements
Question
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also, find the maximum volume.
Advertisements
Solution 1
Let x and y be the length and breadth of a given rectangle ABCD as per question, the rectangle be revolved about side AD which will make a cylinder with radius x and height y.
∴ Volume of the cylinder V = `(pi"r"^2)/"h"`
⇒ V = πx2y ...(i)
Now perimeter of rectangle P = 2(x + y)
⇒ 36 = 2(x + y)
⇒ x + y = 18
⇒ y = 18 – x ...(iii)
Putting the value of y in equation (i) we get
V = πx2(18 – x)
⇒ V = π(18x2 – x3)
Differentiating both sides w.r.t. x, we get
`"dV"/"dx" = π(36x - 3x^2)` ...(iii)
For local maxima and local minima `"dV"/"dx"` = 0
∴ π(36x – 3x2) = 0
⇒ 36x – 3x2 = 0
⇒ 3x(12 – x) = 0
⇒ x ≠ 0
∴ 12 – x = 0
⇒ x = 12
From equation (ii)
y = 18 – 12
y = 6
Differentiating equation (iii) w.r.t. x, we get
`("d"^2"V")/("dx"^2) = pi(36 - 6x)`
At x = 12 `("d"^2"V")/("dx"^2)`
= π(36 – 6 × 12)
= π(36 – 72)
= – 36π < 0 maxima
Now volume of the cylinder so formed = πx2y
= π × (12)2 × 6
= π × 144 × 6
= 864π cm3
Hence, the required dimensions are 12 cm and 6 cm and the maximum volume is 864π cm3.
Solution 2
Let the length and breadth of rectangle be x and y respectively.
Given, P = 36
2(x + y) = 36
x + y = 18
y = 18 – x
Let the rectangle revolved around side x and form cylinder with radius r and height (18 – x)
2πr = x
`r = x/(2π)`
Volume, V = πr2h
= `π(x/(2π))^2y`
= `π x^2/(4π^2) (18 - x)`
= `1/(4π) x^2 (18 - x)`
`V = 1/(4π) (18x^2 - x^3)`
Differentiate w.r.t. x
`(dV)/(dx) = 1/(4π) (36x - 3x^2)`
`(dV)/(dx) = 0`
For maxima and minima,
`1/(4π) (36x - 3x^2) = 0`
x ≠ 0, x = 12
`(d^2V)/(dx^2) = 1/(4π) (36 - 6x)`
`(d^2V)/(dx^2)]_(x = 12) = 1/(4π) [36 - 6(12)]`
= `1/(4π) (-36) < 0`
Hence, Volume is maximum when x = 12
y = 18 – x
= 18 – 12
= 6
Required length and breadth of rectangle are 12 and 6 respectively.
Volume `V = π(12/(2π))^2 (6)`
= `36/22 xx 6 xx 7`
= 68.72 unit3
APPEARS IN
RELATED QUESTIONS
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.
Find the maximum and minimum value, if any, of the following function given by f(x) = |sin 4x + 3|
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) = sinx − cos x, 0 < x < 2π
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 maximum and minimum values of x + sin 2x on [0, 2π].
Find two numbers whose sum is 24 and whose product is as large as possible.
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.`
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 : f(x) = x3 – 9x2 + 24x
Find the maximum and minimum of the following functions : f(x) = `logx/x`
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?
A box with a square base is to have an open top. The surface area of the box is 192 sq cm. What should be its dimensions in order that the volume is largest?
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
Divide the number 20 into two parts such that their product is maximum.
If f(x) = x.log.x then its maximum value is ______.
If x + y = 3 show that the maximum value of x2y is 4.
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
Divide the number 20 into two parts such that their product is maximum
Twenty meters of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is ______
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 curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
The function `f(x) = x^3 - 6x^2 + 9x + 25` has
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
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 ______.
Let P(h, k) be a point on the curve y = x2 + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P is ______.
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
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 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 ______.
A straight line is drawn through the point P(3, 4) meeting the positive direction of coordinate axes at the points A and B. If O is the origin, then minimum area of ΔOAB is equal to ______.
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.
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.
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.
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?
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`
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
