A rectangle is inscribed in an equilateral triangle of side length $$2a$$ units. The maximum area of this rectangle can be
In $$\triangle DBF,\tan { 60° } =\cfrac { 2b }{ 2a-l } $$
$$\sqrt { 3 } =\cfrac { 2b }{ 2a-l } $$
$$b=\cfrac { \sqrt { 3 } \left( 2a-l \right) }{ 2 } $$
A, Area of rectangle $$DEFG=l\times b$$
$$=l\times \cfrac { \sqrt { 3 } }{ 2 } (2a-l)$$
For maximum area
$$\cfrac { dA }{ dl } =0$$
$$\cfrac { \sqrt { 3 } }{ 2 } \left( 2a-2l \right) =0$$
$$a=l$$
Therefore, Maximum Area$$=l\times b$$
$$=a\times \cfrac { \sqrt { 3 } }{ 2 } (2a-a)$$
Maximum Area$$=\cfrac { \sqrt { 3 } { a }^{ 2 } }{ 2 } $$ (Answer)
A helicopter is flying along the curve given by $$y - x^{3/2} =7, (x \ge 0)$$. A solider positioned at the point $$\left(\dfrac{1}{2}, 7\right)$$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is:
A tangent to the curve, $$y=f(x)$$ at $$P(x, y)$$ meets $$x-$$axis at $$A$$ and $$y-$$axis at $$B$$. If $$AP:BP=1:3$$ and $$f(1)=1$$, then the curve also passes through the point.
Match List I with List II and select the correct answer using the code given below the lists: List I List II (A) $$f(x) = \cos x$$ $$1.$$ The graph cuts y-axis in infinite number of points (B) $$f(x) = ln \ x$$ $$2.$$ The graph cut x-axis in two points. (C) $$f(x) = x^{2} - 5x + 4$$ $$3.$$ The graph cuts y-axis in only one point (D) $$f(x) = e^{x}$$ $$4.$$ The graph cuts x-axis in only one point $$5.$$ The graph cuts x-axis in infinite number of points
Let $$f(x)=\dfrac{\sin \pi X}{x^2},x>0$$
Let $$x_1
If the normal to the curve $$y=f(x)$$ at the point $$(3,\,4)$$ make an angle $$\dfrac{3\pi}{4}$$ with the positive x-axis, then $$f'(3)$$ is
The points at which the tangents to the curve $$y=x^3-12x+18$$ are parallel to x-axis are?
Let $$y = f(x)$$ be a parabola, having its axis parallel to y-axis, which is touched by the line $$y =x$$ at $$x = 1$$, then
Radius of the circle that passes through origin and touches the parabola $$y^2 = 4ax$$ at the point $$(a, 2a)$$ is
The volume of the greatest cylinder which can be inscribed in a cone of height $$30cm$$ and semi-vertical angle $$30^0$$ is
Tangents are drawn to $$x^2+y^2=16$$ from the point $$P(0,h)$$. These tangents meet the x-axis at $$A$$ and $$B$$. If the area of $$\Delta PAB$$ is minimum, then $$h=?$$