Advertisements
Advertisements
प्रश्न
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 ?
उत्तर
\[\text { Let l, b and V be the length, breadth and volume of the rectangle, respectively . Then, }\]
\[2\left( l + b \right) = 36\]
\[ \Rightarrow l = 18 - b . . . \left( 1 \right)\]
\[\text { Volume of the cylinder when revolved about the breadth, V } = \pi l^2 b\]
\[ \Rightarrow V = \pi \left( 18 - b \right)^2 b .............\left[\text{From eq. }\left( 1 \right) \right]\]
\[ \Rightarrow V = \pi\left( 324b + b^3 - 36 b^2 \right)\]
\[ \Rightarrow \frac{dV}{db} = \pi\left( 324 + 3 b^2 - 72b \right)\]
\[\text { For the maximum or minimum values of V, we must have }\]
\[\frac{dV}{db} = 0\]
\[ \Rightarrow \pi\left( 324 + 3 b^2 - 72b \right) = 0\]
\[ \Rightarrow 324 + 3 b^2 - 72b = 0\]
\[ \Rightarrow b^2 - 24b + 108 = 0\]
\[ \Rightarrow b^2 - 6b - 18b + 108 = 0\]
\[ \Rightarrow \left( b - 6 \right)\left( b - 18 \right) = 0\]
\[ \Rightarrow b = 6, 18\]
\[\text { Now,} \]
\[\frac{d^2 V}{d b^2} = \pi\left( 6b - 72 \right)\]
\[\text { At }b = 6: \]
\[\frac{d^2 V}{d b^2} = \pi\left( 6 \times 6 - 72 \right)\]
\[ \Rightarrow \frac{d^2 V}{d b^2} = - 36\pi < 0\]
\[\text{ At } b= 18: \]
\[\frac{d^2 V}{d b^2} = \pi\left( 6 \times 18 - 72 \right)\]
\[ \Rightarrow \frac{d^2 V}{d b^2} = 36\pi > 0\]
\[\text { Substitutingthe value of b in eq. } \left( 1 \right),\text { we get }\]
\[l = 18 - 6 = 12\]
\[\text { So, the volume is maximum when l = 12 cm and b = 6 cm }. \]
APPEARS IN
संबंधित प्रश्न
f(x)=| x+2 | on R .
f(x) = \[\frac{1}{x^2 + 2}\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/2`
Find the point of local maximum or local minimum, if any, of the following function, using the first derivative test. Also, find the local maximum or local minimum value, as the case may be:
f(x) = x3(2x \[-\] 1)3.
`f(x) = 2/x - 2/x^2, x>0`
f(x) = xex.
`f(x) = x/2+2/x, x>0 `.
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
`f(x)=xsqrt(1-x), x<=1` .
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
f(x) = (x \[-\] 1)2 + 3 in [ \[-\] 3,1] ?
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
Find the absolute maximum and minimum values of the function of given by \[f(x) = \cos^2 x + \sin x, x \in [0, \pi]\] .
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
How should we choose two numbers, each greater than or equal to `-2, `whose sum______________ so that the sum of the first and the cube of the second is minimum?
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?
Find the point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
Manufacturer can sell x items at a price of rupees \[\left( 5 - \frac{x}{100} \right)\] each. The cost price is Rs \[\left( \frac{x}{5} + 500 \right) .\] Find the number of items he should sell to earn maximum profit.
A particle is moving in a straight line such that its distance at any time t is given by S = \[\frac{t^4}{4} - 2 t^3 + 4 t^2 - 7 .\] Find when its velocity is maximum and acceleration minimum.
Write the minimum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
The maximum value of x1/x, x > 0 is __________ .
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
The number which exceeds its square by the greatest possible quantity is _________________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
The point on the curve y2 = 4x which is nearest to, the point (2,1) is _______________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
The minimum value of x loge x is equal to ____________ .
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?
