Advertisements
Advertisements
Question
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
Solution
\[\text { Let x andybe the length and breadth of the rectangular page, respectively. Then,}\]
\[\text { Area of the page } = 150\]
\[ \Rightarrow xy = 150\]
\[ \Rightarrow y = \frac{150}{x} . . . \left( 1 \right)\]
\[\text { Area of the printed matter }=\left( x - 3 \right)\left( y - 2 \right)\]
\[ \Rightarrow A = xy - 2x - 3y + 6\]
\[ \Rightarrow A = 150 - 2x - \frac{450}{x} + 6\]
\[ \Rightarrow \frac{dA}{dx} = - 2 + \frac{450}{x^2}\]
\[\text { For maximum or minimum values of A, we must have }\]
\[\frac{dA}{dx} = 0\]
\[ \Rightarrow - 2 + \frac{450}{x^2} = 0\]
\[ \Rightarrow 2 x^2 = 450\]
\[ \Rightarrow x = 15\]
\[\text { Substituting the value of x in }\left( 1 \right), \text { we get }\]
\[y = 10\]
\[\text { Now }, \]
\[\frac{d^2 A}{d x^2} = \frac{- 900}{x^3}\]
\[ \Rightarrow \frac{d^2 A}{d x^2} = \frac{- 900}{\left( 15 \right)^3}\]
\[ \Rightarrow \frac{d^2 A}{d x^2} = \frac{- 900}{3375} < 0\]
\[\text { So, area of the printed matter is maximum when x = 15 and y = 10 } . \]
APPEARS IN
RELATED QUESTIONS
f(x) = | sin 4x+3 | on R ?
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = 16x2 \[-\] 16x + 28 on R ?
f(x) = (x \[-\] 5)4.
f(x) = (x \[-\] 1) (x+2)2.
f(x) = cos x, 0 < x < \[\pi\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/2`
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) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
f(x) = xex.
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the we should be cut so that the sum of the areas of the square and triangle is minimum?
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.
A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?
Show that among all positive numbers x and y with x2 + y2 =r2, the sum x+y is largest when x=y=r \[\sqrt{2}\] .
Find the point on the curvey y2 = 2x which is at a minimum distance from the point (1, 4).
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
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.
Write sufficient conditions for a point x = c to be a point of local maximum.
Write the point where f(x) = x log, x attains minimum value.
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
For the function f(x) = \[x + \frac{1}{x}\]
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
f(x) = \[\sin + \sqrt{3} \cos x\] is maximum when x = ___________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .
f(x) = 1+2 sin x+3 cos2x, `0<=x<=(2pi)/3` is ________________ .
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
