Advertisements
Advertisements
Question
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.
Advertisements
Solution
Let the side of the square cut off from the corners be x cm.
Therefore, each side of the square box is (18 – 2x) cms and the height is x cms.
Let V be the volume of the box.
V = Area of the base × Height
V = (18 − 2x)2x
V = (324 − 72x + 4x2) x
∴ V = 4x3 − 72x2 + 324x
Differentiating w.r.t. x, we get
`(dV)/dx = 12x^2 - 144x + 324`
`therefore (d^2V)/dx^2 = 24x - 144`
For maximum volume, `(dV)/dx = 0`
∴ 12x2 − 144x + 324 = 0
∴ x2 − 12x + 27 = 0
∴ (x − 3) (x − 9) = 0
∴ x − 3 = 0 or x − 9 = 0
∴ x = 3 or x = 9
But x ≠ 9
∴ x = 3
For x = 3
`((d^2V)/dx^2)_(x = 3) = 24(3) - 144 = -72 < 0`
The volume of the box is maximum when x = 3.
∴ Maximum value of the box = (18 − 6)2 (3)
= 432 cc
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.
A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.
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:
g(x) = x3 − 3x
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
Prove that the following function do not have maxima or minima:
f(x) = ex
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
Find two numbers whose sum is 24 and whose product is as large as possible.
Find the maximum and minimum of the following functions : f(x) = x log x
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.
Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
Determine the maximum and minimum value of the following function.
f(x) = x log x
If f(x) = x.log.x then its maximum value is ______.
If x + y = 3 show that the maximum value of x2y is 4.
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
Divide the number 20 into two parts such that their product is maximum
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 value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
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 ______.
If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is `pi/3`
Find the points of local maxima, local minima and the points of inflection of the function f(x) = x5 – 5x4 + 5x3 – 1. Also find the corresponding local maximum and local minimum values.
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?
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
The maximum value of `(1/x)^x` is ______.
The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
If y = x3 + x2 + x + 1, then y ____________.
Find the area of the largest isosceles triangle having a perimeter of 18 meters.
Divide 20 into two ports, so that their product is maximum.
If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
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.
If x + y = 8, then the maximum value of x2y is ______.
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?
Sumit has bought a closed cylindrical dustbin. The radius of the dustbin is ‘r' cm and height is 'h’ cm. It has a volume of 20π cm3.
- Express ‘h’ in terms of ‘r’, using the given volume.
- Prove that the total surface area of the dustbin is `2πr^2 + (40π)/r`
- Sumit wants to paint the dustbin. The cost of painting the base and top of the dustbin is ₹ 2 per cm2 and the cost of painting the curved side is ₹ 25 per cm2. Find the total cost in terms of ‘r’, for painting the outer surface of the dustbin including the base and top.
- Calculate the minimum cost for painting the dustbin.
The absolute maximum value of the function f(x) = 2x3 − 3x2 − 36x + 9 defined on [−3, 3] is ______.
