Advertisements
Advertisements
प्रश्न
The total area of a page is 150 cm2. The combined width of the margin at the top and bottom is 3 cm and the side 2 cm. What must be the dimensions of the page in order that the area of the printed matter may be maximum?
Advertisements
उत्तर
\[\text { Let x andybe the length and breadth of the rectangular page, respectively. Then,}\]
\[\text { Area of the page } = 150\]
\[ \Rightarrow xy = 150\]
\[ \Rightarrow y = \frac{150}{x} . . . \left( 1 \right)\]
\[\text { Area of the printed matter }=\left( x - 3 \right)\left( y - 2 \right)\]
\[ \Rightarrow A = xy - 2x - 3y + 6\]
\[ \Rightarrow A = 150 - 2x - \frac{450}{x} + 6\]
\[ \Rightarrow \frac{dA}{dx} = - 2 + \frac{450}{x^2}\]
\[\text { For maximum or minimum values of A, we must have }\]
\[\frac{dA}{dx} = 0\]
\[ \Rightarrow - 2 + \frac{450}{x^2} = 0\]
\[ \Rightarrow 2 x^2 = 450\]
\[ \Rightarrow x = 15\]
\[\text { Substituting the value of x in }\left( 1 \right), \text { we get }\]
\[y = 10\]
\[\text { Now }, \]
\[\frac{d^2 A}{d x^2} = \frac{- 900}{x^3}\]
\[ \Rightarrow \frac{d^2 A}{d x^2} = \frac{- 900}{\left( 15 \right)^3}\]
\[ \Rightarrow \frac{d^2 A}{d x^2} = \frac{- 900}{3375} < 0\]
\[\text { So, area of the printed matter is maximum when x = 15 and y = 10 } . \]
APPEARS IN
संबंधित प्रश्न
f(x)=sin 2x+5 on R .
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = sin 2x, 0 < x < \[\pi\] .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
f(x) = cos x, 0 < x < \[\pi\] .
f(x) =\[x\sqrt{1 - x} , x > 0\].
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) = x4 \[-\] 62x2 + 120x + 9.
f(x) = x3\[-\] 6x2 + 9x + 15
`f(x) = 2/x - 2/x^2, x>0`
f(x) = xex.
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
f(x) = (x \[-\] 1) (x \[-\] 2)2.
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
f(x) = (x \[-\] 1)2 + 3 in [ \[-\] 3,1] ?
Find the maximum value of 2x3\[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].
Find the absolute maximum and minimum values of a function f given by f(x) = 2x3 − 15x2 + 36x + 1 on the interval [1, 5].
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
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 \[M = \frac{WL}{2}x - \frac{W}{2} x^2\] .
Find the point at which M is maximum in a given case.
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?
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 point on the curvey y2 = 2x which is at a minimum distance from the point (1, 4).
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.
If f(x) attains a local minimum at x = c, then write the values of `f' (c)` and `f'' (c)`.
Find the least value of f(x) = \[ax + \frac{b}{x}\], where a > 0, b > 0 and x > 0 .
Write the maximum value of f(x) = x1/x.
Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .
For the function f(x) = \[x + \frac{1}{x}\]
The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
If x+y=8, then the maximum value of xy is ____________ .
f(x) = \[\sin + \sqrt{3} \cos x\] is maximum when x = ___________ .
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
The function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x = ________________ .
Let f(x) = 2x3\[-\] 3x2\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .
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?
Which of the following graph represents the extreme value:-
