Ex. 5. The length of a rectangle is twice its breadth. If its length is decreased by 5 cm and breadth is increased by 5 cm, the area of the rectangle is increased by 75 sq. cm. Find the length of the rectangle.
Ex. 6. In measuring the sides of a rectangle, one side is taken 5% in excess, and the other 4% in deficit. Find the error percent in the area calculated from these measurements.(M.B.A. 2003)
Sol. Let x and y be the sides of the rectangle. Then, Correct area = xy.
Calculated area = (105/100)*x * (96/100)*y = (504/500 )(xy)
Error In measurement = (504/500)xy- xy = (4/500)xy
Ex. 7. A rectangular grassy plot 110 m. by 65 m has a gravel path 2.5 m wide all round it on the inside. Find the cost of gravelling the path at 80 paise per sq. metre.
Sol. Area of the plot = (110 x 65) m2 = 7150 m2
Area of the plot excluding the path = [(110 - 5) * (65 - 5)] m2 = 6300 m2.
Area of the path = (7150 - 6300) m2 = 850 m2.
Cost of gravelling the path = Rs.850 * (80/100)= Rs. 680
Ex. 8. The perimeters of two squares are 40 cm and 32 cm. Find the perimeter ofa third square whose area is equal to the difference of the areas of the two squares. (S.S.C. 2003)
Sol. Side of first square = (40/4) = 10 cm;
Side of second square = (32/4)cm = 8 cm.
Area of third square = [(10) 2 - (8) 2] cm2 = (100 - 64) cm2 = 36 cm2.
Side of third square = (36)(1/2) cm = 6 cm.
Required perimeter = (6 x 4) cm = 24 cm.
Ex. 9. A room 5m 55cm long and 3m 74 cm broad is to be paved with square tiles. Find the least number of square tiles required to cover the floor.
Sol. Area of the room = (544 x 374) cm2.
Size of largest square tile = H.C.F. of 544 cm and 374 cm = 34 cm.
Area of 1 tile = (34 x 34) cm2.
Number of tiles required =(544*374)/(34*34)=176
Ex. 10. Find the area of a square, one of whose diagonals is 3.8 m long.
Sol. Area of the square = (1/2)* (diagonal) 2 = [(1/2)*3.8*3.8 ]m2 = 7.22 m2.
Ex. 11. The diagonals of two squares are in the ratio of 2 : 5. Find the ratio of their areas. (Section Officers', 2003)
Sol. Let the diagonals of the squares be 2x and 5x respectively.
Ratio of their areas = (1/2)*(2x) 2:(1/2)*(5x) 2= 4x2: 25x2= 4 : 25.
Ex.12. If each side of a square is increased by 25%, find the percentage change in its area.
Sol. Let each side of the square be a. Then, area = a2.
New side =(125a/100) =(5a/4). New area = (5a/4) 2 =(25a2)/16.
Increase in area = ((25 a2)/16)-a2=(9a2)/16.
Increase% = [((9a2)/16)*(1/a2)*100] % = 56.25%.
Ex. 13. If the length of a certain rectangle is decreased by 4 cm and the width is increased by 3 cm, a square with the same area as the original rectangle would result. Find the perimeter of the original rectangle.
Sol.Let x and y be the length and breadth of the rectangle respectively.
Then, x - 4 = y + 3 or x - y = 7----(i)
Area of the rectangle =xy; Area of the square = (x - 4) (y + 3)
(x - 4) (y + 3) =xy <=> 3x - 4y = 12----(ii)
Solving (i) and (ii), we get x = 16 and y = 9.
Perimeter of the rectangle = 2 (x + y) = [2 (16 + 9)] cm = 50 cm.
Ex. 14. A room is half as long again as it is broad. The cost of carpeting the at Rs. 5 per sq. m is Rs. 270 and the cost of papering the four walls at Rs. 10 per m2 is Rs. 1720. If a door and 2 windows occupy 8 sq. m, find the dimensions of the room.
Sol. Let breadth = x metres, length = 3x metres, height = H metres.
Area of the floor=(Total cost of carpeting)/(Rate/m2)=(270/5)m2=54m2.
x* (3x/2) = 54 <=> x^2 = (54*2/3) = 36 <=> x = 6.
So, breadth = 6 m and length =(3/2)*6 = 9 m.
Now, papered area = (1720/10)m2 = 172 m2.
Area of 1 door and 2 windows = 8 m2.
Total area of 4 walls = (172 + 8) m2 = 180 m2
2*(9+ 6)* H = 180 <=> H = 180/30 = 6 m.
Ex. 15. Find the area of a triangle whose sides measure 13 cm, 14 cm and 15 cm.
Sol. Let a = 13, b = 14 and c = 15. Then, S =(1/2)(a + b + c) = 21.
(s- a) = 8, (s - b) = 7 and (s - c) = 6.
Area = (s(s- a) (s - b)(s - c))(1/2) = (21 *8 * 7*6)(1/2) = 84 cm2.
Ex. 16. Find the area of a right-angled triangle whose base is 12 cm and hypotenuse is 13cm.
Sol. Height of the triangle = [(13) 2- (12) 2](1/2) cm = (25)(1/2) cm = 5 cm.
Its area = (1/2)* Base * Height = ((1/2)*12 * 5) cm2 = 30 cm2.
Ex. 17. The base of a triangular field is three times its altitude. If the cost of cultivating the field at Rs. 24.68 per hectare be Rs. 333.18, find its base and height.
Sol. Area of the field = Total cost/rate = (333.18/25.6)hectares = 13.5 hectares
Area of rhombus = (1/2) x (Product of diagonals) = ((1/2)x 24 x 32) cm^2 = 384 cm^2
Ex. 23. The difference between two parallel sides of a trapezium is 4 cm. perpendicular distance between them is 19 cm. If the area of the trapezium is 475 find the lengths of the parallel sides.(R.R.B. 2002)
Sol. Let the two parallel sides of the trapezium be a em and b em.
Then, a - b = 4
And, (1/2) x (a + b) x 19 = 475ó(a + b) =((475 x 2)/19) óa + b = 50
Solving (i) and (ii), we get: a = 27, b = 23.
So, the two parallel sides are 27 cm and 23 cm.
Ex. 24. Find the length of a rope by which a cow must be tethered in order tbat it
may be able to graze an area of 9856 sq. metres.(M.A.T. 2003)
Sol. Clearly, the cow will graze a circular field of area 9856 sq. metres and radiusequal to the length of the rope.
Let the length of the rope be R metres.
Then, Õ(R)^2 = (9856 X (7/22)) = 3136ó R = 56.
Length of the rope = 56 m.
Ex. 25. The area of a circular field is 13.86 hectares. Find the cost of fencing it at
the rate of Rs. 4.40 per metre.
Sol. Area = (13.86 x 10000) m2= 138600 m2.
Õ(R2= 138600ó(R)2 = (138600 x (7/22)) ó R = 210 m.
Circumference = 2ÕR = (2 x (22/7) x 210) m = 1320 m.
Cost of fencing = Rs. (1320 x 4.40) = Rs. 5808.
Ex. 26. The diameter of the driving wheel of a bus is 140 em. How many revolution, per minute must the wheel make in order to keep a speed of 66 kmph ?
Sol. Distance to be covered in 1 min. = (66 X_1000)/(60) m = 1100 m.
Circumference of the wheel = (2 x (22/7) x 0.70) m = 4.4 m.
Number of revolutions per min. =(1100/4.4) = 250.
Ex, 27. A wheel makes 1000 revolutions in covering a distance of 88 km. Find the radius of the wheel.
Sol. Distance covered in one revolution =((88 X 1000)/1000)= 88m.
2ÕR = 88 ó 2 x (22/7) x R = 88 óR = 88 x (7/44) = 14 m.
Ex, 28. The inner circumference of a circular race track, 14 m wide, is 440 m. Find
radius of the outer circle.
Sol . Let inner radius be r metres. Then, 2Õr= 440 ó r = (440 x (7/44))= 70 m.
Radius of outer circle = (70 + 14) m = 84 m.
Ex, 29. Two concentric circles form a ring. The inner and outer circumferences of ring are (352/7) m and (518/7) m respectively. Find the width of the ring.
Sol.. Let the inner and outer radii be r and R metres.
Then 2Õr = (352/7) ó r =((352/7) X (7/22) X (1/2))=8m.
2ÕR=(528/7) ó R=((528/7) X (7/22) X (1/2))= 12m.
, ',Width of the ring = (R - r) = (12 - 8) m = 4 m.
Ex, 30. A sector of 120', cut out from a circle, has an area of (66/7) sq. cm. Find the
radius of the circle.
Sol. Let the radius of the circle be r cm. Then,
( Õ( r )2 q) /360=(66/7) ó (22/7) X ( r ) 2 X(120/360)= (66/7)
ó ( r )2=((66/7) X (7/22) X 3) ó r=3.
Hence, radius = 3 cm.
Ex, 31. Find the ratio of the areas of the incircle and circumcircle of a square.
Sol. Let the side of the square be x. Then, its diagonal = Ö2 x.
Radius of incircle = (x/2)
Radius of circum circle= (Ö2x/2) =(x/Ö2)
Required ratio = ((Õ ( r )2 )/4 : (Õ( r )2) /2) = (1/4) : 1/2) = 1 : 2.
Ex. 32. If the radius of a circle is decreased by 50%, find the percentage decrease
in its area.
Sol. Let original radius = R. New radius =(50/100)R = (R/2)
Original area=Õ( R )2= and new area= Õ ((R/2))2= (Õ ( R )2)/4
Decrease in area =((3Õ (R )2 )/4 X (1/Õ( R)2) X 100) % = 75%