Advertisements
Advertisements
Question
A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.
Advertisements
Solution
Let the side of the square to be cut off be x cm.
Then, the length and the breadth of the box will be (18 − 2x) cm each and height of the box will be x cm.
Volume of the box, V(x) = x(18 − 2x)2
\[V'\left( x \right) = \left( 18 - 2x \right)^2 - 4x\left( 18 - 2x \right)\]
\[ = \left( 18 - 2x \right)\left( 18 - 2x - 4x \right)\]
\[ = \left( 18 - 2x \right)\left( 18 - 6x \right)\]
\[ = 12\left( 9 - x \right)\left( 3 - x \right)\]
\[V''\left( x \right) = 12\left( - \left( 9 - x \right) - \left( 3 - x \right) \right)\]
\[ = - 12\left( 9 - x + 3 - x \right)\]
\[ = - 24\left( 6 - x \right)\]
\[\text { For maximum and minimum values of V, we must have }\]
\[ V'\left( x \right) = 0\]
\[\Rightarrow\] x = 9 or x = 3
If x = 9, then length and breadth will become 0.
∴ x ≠ 9
\[\Rightarrow\] x = 3
Now,
\[V''\left( 3 \right) = - 24\left( 6 - 3 \right) = - 72 < 0\]
∴ x = 3 is the point of maxima.
\[V\left( x \right) = 3 \left( 18 - 6 \right)^2 = 3 \times 144 = 432 {cm}^3\]
Hence, if we remove a square of side 3 cm from each corner of the square tin and make a box from the remaining sheet, then the volume of the box so obtained would be the largest, i.e. 432 cm3
APPEARS IN
RELATED QUESTIONS
f(x) = 4x2 + 4 on R .
f(x) = - (x-1)2+2 on R ?
f(x)=| x+2 | on R .
f(x) = x3 (x \[-\] 1)2 .
f(x) = sin 2x, 0 < x < \[\pi\] .
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
f(x) = x4 \[-\] 62x2 + 120x + 9.
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by \[M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}\] .
Find the point at which M is maximum in a given case.
Given the sum of the perimeters of a square and a circle, show that the sum of there areas is least when one side of the square is equal to diameter of the circle.
Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.
Two sides of a triangle have lengths 'a' and 'b' and the angle between them is \[\theta\]. What value of \[\theta\] will maximize the area of the triangle? Find the maximum area of the triangle also.
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible?
A tank with rectangular base and rectangular sides, open at the top, is to the constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost 70 per square metre for the base and Rs 45 per square metre for sides, what is the cost of least expensive tank?
A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10 m. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.
A large window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 metres find the dimensions of the rectangle will produce the largest area of the window.
Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]
An isosceles triangle of vertical angle 2 \[\theta\] is inscribed in a circle of radius a. Show that the area of the triangle is maximum when \[\theta\] = \[\frac{\pi}{6}\] .
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r.
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 ?
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
The total cost of producing x radio sets per day is Rs \[\left( \frac{x^2}{4} + 35x + 25 \right)\] and the price per set at which they may be sold is Rs. \[\left( 50 - \frac{x}{2} \right) .\] Find the daily output to maximum the total profit.
The space s described in time t by a particle moving in a straight line is given by S = \[t5 - 40 t^3 + 30 t^2 + 80t - 250 .\] Find the minimum value of acceleration.
Write necessary condition for a point x = c to be an extreme point of the function f(x).
If f(x) attains a local minimum at x = c, then write the values of `f' (c)` and `f'' (c)`.
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .
A wire of length 34 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a rectangle whose length is twice its breadth. What should be the lengths of the two pieces, so that the combined area of the square and the rectangle is minimum?
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
