MATHS WORKSHOP NTSE MENSURATION (SURFACE AREA AND VOLUME) PAGE#
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VOLUME AND SURFACE AREA OF A FRUSTUM OF A RIGHT CIRCULAR CONE Total surface area of the frustum = curved surface area + surface area of circular bases = (r + r) l + r + r If h be the height & l be slant height of the frustum and r & r (r > r) be radii of the two ends then (a) Base area : Top base area = r Bottom base area = r V (b) Slant height of the frustum, l = h ( r r ) (c) (d) Curved surface area Total surface area (e) = l (r+r) = l (r + r) + r + r = [l (r + r) + r + r ] Volume of the frustum = h (r + r + rr ) 3 h-h l l A r A h B r O h B CONVERSION OF SOLID FROM ONE SHAPE TO ANOTHER Some solids like candle, clay etc. can be changed into any shape. But the volume of the both solid shapes are same. For example, if a candle which is generally in the shape of a cylinder can be changed into any shape, but the volume remains same. If a solid is transformed into a number of small identical solids of same or a different shape, then Number of small items = Volume of larger object Volume of a smaller object PIPES AND CANALS Volume of water released by a pipe/canal = Area of cross-section Rate of flow Time In case of a pipe, the cross-section is usually a circle. In case of a canal, the cross-section is usually a rectangle or a trapezium. If rate of flow is given in km/hr, it can be converted into m/sec by multiplying with 5. 8 If time is given in minutes or hours, it must be converted into seconds because rate of flow is in m/sec. Volume of water standing in a field =Area of field Height of standing water If area of field is given in hectares, it must be converted into m by multiplying with 0,000. Height of standing water must be taken in metres. If given in cm it must be converted to metre bydividing by 00. PAGE# 3
EXERCISE Q. In figure, ABCD is a trapezium in which AB DC; AB = 7 cm; AD = BC = 5 cm and the distance between AB and DC is cm. Find the length of DC and hence, find the area of trapezium ABCD. () 30 cm () 0 cm (3) 60 cm () 50 cm Q. Area of a trapezium is 9 cm and its height is 7 cm. If one of the parallel sides is longer than the other side by 8 cm, the length (in cm) of the smaller side is : () 6 () 7 (3) 9 () 7 Q.3 Find the area of the shaded region if every circle is of unit radius. () 3 Sq. Units () 3 Sq.Units (3) 3 Sq.Units Q. () cm (3) cm () None of these In the adjoining figure, the radius of the inner circle, if other circles are of radii m, is : () (3) Q.6 3 Sq.Units The area of a circle inscribed in an equilateral triangle is 5 cm. Find the perimeter of the triangle. () 3 cm Q.5 () m m () m () m An arch in the form of a circle has a span of 30 meters and a height of 0 meters. The radius of the arch in meters is ().5 () 5.50 (3) 6.75 () 6.5 PAGE#
Q.7 ABCP is a quadrant of a circle of radius cm. With AC as diameter a semi circle is drawn (figure). Find the perimeter and area of the shaded region. () 00 cm Q.8 () 9 cm (3) 96 cm () 98 cm Two squares have dimensions as indicated in the drawing. What is the area of the shaded region? 5cm () ½ sq. cm Q.9 () 3½ sq. cm (3) 5 sq. cm () 7 sq. cm In the given fig. X andy are centres of circles of radii 9 cm and cm respectively such that XY= 7 cm. Also Z is the centre of a circle of radius r cm, which touches the above circles externally such that XZY = 90º. Measurement of r is : Z r cm cm 9 r Y X 7 cm () 6 cm () 5.5 cm (3) 6.5 cm () 6.5 cm Q.0 The surface of water in a swimming pool is a rectangle 3 m long and 9.5 m wide and the depth of the water increases uniformly from.5 m at one end to.5 m at the other end. What is the volume of the water in the pool (in cubic metre)? () 00 () 060 (3) 06 () 065 Q. Water in a canal, 30 dm wide and dm deep, is flowing with a speed of 0 km/hour. How much area will it irrigate in 30 minutes, if 8 cm of standing water is required for irrigation. () 500 m () 05000 m (3) 00000 m () 5000 m PAGE# 5
Q. The dimensions of a hall are 90 m, 0 m and 6 m. How many persons can sit in the hall, if each required 0 m3 of air? () 50 persons () 80 persons (3) 50 () 60 Q.3 If the radius of a right circular cylinder is increased by 50% and height is decreased by 0% then the percentage change in volume of cylinder is () 0% () 50% (3) 60% () 80% Q. A cone, a right cylinder and a hemi shpere stand on equal bases and have the same height. Their volumes are in the ratio () : : 3 () : 3 : (3) : 3 : () : : 3 Q.5 The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be () 0 cm 7 of the volume of the given cone, at what height above the base is the section made? () 0 cm. (3) 30 cm () 5 cm Q.6 A hemispherical bowl of internal diameter 30 cm contains some liquid. This liquid is to be filled into cylindrical shaped bottles each of diameter 5 cm and height 6 cm. Find the number of bottles necessary to empty the bowl. () 50 bottles () 75 bottles (3) 65 bottles () 60 bottles Q.7 There is a cylinder circumscribing the hemisphere such that their bases are common. The ratio of the volumes of hemisphere to cylinder will be () : 3 () : (3) : 3 () 3 : Q.8 A hemi-spherical depression is cutout from one face of the cubical wooden block such that the diameter of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid. () [ + ) sq. units. () [ + ) sq. units. (3) [ ) sq. units. () [ ) sq. units. Q.9 Determine the ratio of the volume of a cube to that of a sphere which will exactly fit inside the cube. () 6 : () : 6 (3) : 3 () 3 : Q.0 A solid cylinder of diameter cm and height 5 cm is melted and recast into toys in the shape of a right circular cone mounted on a hemisphere. Find the radius of the hemisphere and the total height of the toy if height of the conical part is 3 times its radius. () 3 cm () 9 cm (3) 6 cm () cm PAGE# 6
ANSWER KEY Q. Q.8 Q.5 Q. Q.9 Q.6 3 Q.3 3 Q.0 3 Q.7 3 Q. Q. Q.8 Q.5 Q. Q.9 Q.6 Q.3 Q.0 Q.7 Q. PAGE# 7