Advertisements
Advertisements
Question
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
Solution
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.
RELATED QUESTIONS
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.
Find the maximum and minimum value, if any, of the following function given by f(x) = −(x − 1)2 + 10
Find the maximum and minimum value, if any, of the function given by f(x) = |x + 2| − 1.
Find the maximum and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.
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 functions. Find also the local maximum and the local minimum values, as the case may be:
`f(x) = xsqrt(1-x), x > 0`
Prove that the following function do not have maxima or minima:
g(x) = logx
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) =x^3, x in [-2,2]`
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = sin x + cos x , x ∈ [0, π]
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 their sum is 35 and the product x2y5 is a maximum.
Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has
- local maxima
- local minima
- point of inflexion
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
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.
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.
The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is Rs. 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it.
Find the point on the straight line 2x+3y = 6, which is closest to the origin.
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.
An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of `pia^3`cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
Determine the maximum and minimum value of the following function.
f(x) = x log x
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?
If x + y = 3 show that the maximum value of x2y is 4.
The function f(x) = x log x is minimum at 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?
By completing the following activity, examine the function f(x) = x3 – 9x2 + 24x for maxima and minima
Solution: f(x) = x3 – 9x2 + 24x
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme values, f'(x) = 0, we get
x = `square` or `square`
∴ f''`(square)` = – 6 < 0
∴ f(x) is maximum at x = 2.
∴ Maximum value = `square`
∴ f''`(square)` = 6 > 0
∴ f(x) is maximum at x = 4.
∴ Minimum value = `square`
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
Find both the maximum and minimum values respectively of 3x4 - 8x3 + 12x2 - 48x + 1 on the interval [1, 4].
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 distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
The maximum value of the function f(x) = `logx/x` is ______.
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 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 ______.
If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
If the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.
A cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere is ______.
The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.
The minimum value of the function f(x) = xlogx is ______.
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
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.
Divide the number 100 into two parts so that the sum of their squares is minimum.
The absolute maximum value of the function f(x) = 2x3 − 3x2 − 36x + 9 defined on [−3, 3] is ______.
