Advertisements
Advertisements
प्रश्न
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?
Advertisements
उत्तर
Let x cm be the side of square base and h cm be its height.
Then x2 + 4xh = 192
∴ h = `(192 - x^2)/(4x)` ...(1)
Let V = `x^2"h"`
= `x^2((192 - x^2)/(4x))` ...[By (1)]
∴ V = `(1)/(4)(192x - x^3)`
∴ `"dV"/("d"x) = (1)/(4) "d"/("d"x)(192x - x^3)`
= `(1)/(4)(192 xx 1 - 3x^2)`
= `(3)/(4)(64 - x^2)`
and
`("d"^2"V")/("d"x^2) - (3"d")/(4"d"x)(64 - x^2)`
= `(3)/(4)(0 - 2x)`
= `-(3)/(2)x`
For maximum V, `"dV"/("d"x)` = 0
∴ `(3)/(4)(64 - x^2)` = 0
∴ x2 = 64
∴ x = 8 ...[∵ x > 0]
and
`(("d"^2V)/("d"x^2))_("at" x = 8)`
= `-(3)/(2) xx 8`
= – 12 < 0
∴ by the second derivative test, V is maximum at x = 8.
If x = 8,
h = `(192 - 64)/(4(8)`
= `(128)/(32)`
= 4
Hence, the volume of the box is largest, when the side of square base is 8 cm and its height is 4 cm.
संबंधित प्रश्न
Find the maximum and minimum value, if any, of the following function given by g(x) = x3 + 1.
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:
`h(x) = sinx + cosx, 0 < x < pi/2`
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:
f(x) = x3 − 6x2 + 9x + 15
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) = x/2 + 2/x, x > 0`
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
`f(x) = xsqrt(1-x), x > 0`
What is the maximum value of the function sin x + cos x?
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.
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.`
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 rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get the maximum area. Also, find the maximum area.
Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`
Find the maximum and minimum of the following functions : f(x) = `logx/x`
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.
Choose the correct option from the given alternatives :
If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
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 ______
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 function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
The maximum value of sin x . cos x is ______.
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
The function `"f"("x") = "x" + 4/"x"` has ____________.
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
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 ______.
If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
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)
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
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
The absolute maximum value of the function f(x) = 2x3 − 3x2 − 36x + 9 defined on [−3, 3] is ______.
For a function defined on an interval \[I\], which condition means that \[f\] has a maximum value at \[c\in I\]?
Assume \[f'(c)=0\] and the second derivative exists at \[c\]. Which condition gives a local maximum?
What are the critical points obtained from \[12x(x-1)(x+2)=0\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what is \[f''(x)\]?
