differentiating the equation on both sides with respect to x, we get,
Find dy/dx in each of the following: $y^{3}-3 x y^{2}=x^{3}+3 x^{2} y$
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Exercise 11.4
On a square cardboard sheet of area \[784\] \[c{{m}^{2}}\], four congruent circular plates of maximum size are placed such that each circular plate touches the other two plates and each side of the square sheet is tangent to two circular plates. Find the area of the square sheet not covered by the circular plates.
Given Area of the square = \[784\] \[c{{m}^{2}}\] Hence Side of the square = \[\sqrt{Area}\] = \[\sqrt{784}\] = \[28\] cm Given that the four circular plates are congruent, Therefore diameter of...
Four circular cardboard pieces of radii \[7\] cm are placed on a paper in such a way that each piece touches other two pieces. Find the area of the portion enclosed between these pieces.
solution From the given information, it is given that the four circles are placed such that each piece touches the other two pieces. Now by joining the centers of the circles by a line segment, we...
Find the area of the sector of a circle of radius \[5\] cm, if the corresponding arc length is \[3.5\] cm.
solution Given Radius of the circle = r = \[5\] cm Given Arc length of the sector = l = \[3.5\] cm Let us consider the central angle (in radians) be \[\theta \]. As we know that Arc length = Radius...
Three circles each of radius \[3.5\] cm are drawn in such a way that each of them touches the other two. Find the area enclosed between these circles.
Solution: Given that the three circles are drawn such that each of them touches the other two. Now, by joining the centers of the three circles, We get, AB = BC = CA = \[2\] (radius) = \[7\] cm...
In Fig. 11.17, ABCD is a trapezium with AB || DC, AB = \[18\] cm, DC = \[32\] cm and distance between AB and DC = \[14\] cm. If arcs of equal radii \[7\] cm with centres A, B, C and D have been drawn, then find the area of the shaded region of the figure.
Solution Given AB = \[18\] cm, DC = \[32\] cm Given, Distance between AB and DC = Height = \[14\] cm We know that Area of the trapezium = (\[1/2\]) × (Sum of parallel sides) × Height =...
A circular pond is \[17.5\] m is of diameter. It is surrounded by a \[2\] m wide path. Find the cost of constructing the path at the rate of Rs \[25\] per \[{{m}^{2}}\]
Solution: Given Diameter of the circular pond = \[17.5\] m Let us consider r be the radius of the park = \[(17.5/2)\] m = \[8.75\] m Given The circular pond is surrounded by a path of width \[2\] m....
Find the area of the segment of a circle of radius \[12\] m whose corresponding sector has a central angle of \[{{60}^{\circ }}\] (Use \[\pi =3.14\]).
Solution: From the given information, Radius of the circle = r = \[12\] cm ∴ OA = OB = \[12\] cm \[\angle AOB={{60}^{\circ }}\] (given) As triangle OAB is an isosceles triangle, ∴ \[\angle...
Find the area of the segment of a circle of radius \[12\] m whose corresponding sector has a central angle of \[{{60}^{\circ }}\] (Use \[\pi =3.14\]).
Solution: From the given information, Radius of the circle = r = \[12\] cm ∴ OA = OB = \[12\] cm \[\angle AOB={{60}^{\circ }}\] (given) As triangle OAB is an isosceles triangle, ∴ \[\angle...
Sides of a triangular field are \[15\] m, \[16\] m and \[17\] m. With the three corners of the field a cow, a buffalo and a horse are tied separately with ropes of length \[7\] m each to graze in the field. Find the area of the field which cannot be grazed by the three animals.
Solution From the given question, We got Sides of the triangle are \[15\] m, \[16\] m and \[17\] m. Then, perimeter of the triangle = \[(15+16+17)\] m = \[48\]m Therefore, Semi-perimeter of the...
The diameters of front and rear wheels of a tractor are \[80\]cm and \[2\] m respectively. Find the number of revolutions that rear wheel will make in covering a distance in which the front wheel makes \[1400\] revolutions.
solution From the given question, We got, Diameter of front wheels = \[{{d}_{1}}\]= \[80\] cm we got, Diameter of rear wheels = \[{{d}_{2}}\]= \[2\]m = \[200\] cm Let us consider \[{{r}_{1}}\] be...
The area of a circular playground is \[22176\] \[{{m}^{2}}\]. Find the cost of fencing this ground at the rate of Rs \[50\] per metre.
From the given question, We got Area of the circular playground = \[22176\] \[{{m}^{2}}\] Let us consider r as the radius of the circle. Therefore, \[\pi {{r}^{2}}=22176\]...