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The length and the breadth of a rectangular door are increased by 1 m each and due to this the area of the door increased by 21 sq. m. But if the length is increased by 1 m and breadth decreased by 1 m, area is decreased by 5 sq. m. Find the perimeter of the door.
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Explanation:
Let original length = l, breadth = b, so area = lb
When l and b increased by 1:
(l+1)(b+1) = lb + 21
Solve, l + b = 20
When l increased by 1, b decreased by 1:
(l+1)(b-1) = lb – 5
Solve, l – b = 6
Now solve both equations, l = 13, b = 7
Perimeter = 2(13+7)
The perimeter of a rectangular plot is 340 m. Find the cost of gardening 1 m broad boundary around it at the rate of Rs 10 per sq. m.
Given 2(l+b) = 340
1 m broad boundary means increase in l and b by 2 m
So area of the boundary will be [(l+2)(b+2) – lb] = 2(l+b) + 4 = 340 + 4 = 344
So cost of gardening = 344*10
The sides of a triangle are in the ratio 3 : 4 : 5 whose area is 216 sq. cm. What will be the perimeter of this triangle?
Sides 3x, 4x, 5x
So semi-perimeter, s = (3x+4x+5x)/2 = 6x
Area = √s(s-a)(s-b)(s-c)
= √6x*3x*2x*x = 6x2 cm2
So 6x2 = 216, this gives x = 6
Perimeter = 12x = 12*6
If the base of a triangle is increased by 50% and its height is decreased by 50%, then what will be the effect on its area?
Area of triangle = (1/2) * base * height
So effect on area = +50 + (-50) + (50)(-50)/100 = -25%
A rectangle whose sides are in the ratio 6 : 5 is formed by bending a circular wire of radius 42 cm. Find the largest side of the rectangle.
Length of wire = 2ᴨr = 264 which should be equal to the perimeter of rectangle in which it is bent.
So 2(6x + 5x) = 264
Solve, x= 12
Largest side = 6x = 6*12
A rectangular sheet of 0.5 cm thickness is made from an iron cube of side 10 cm by hammering it down. The sides of the sheet are in the ratio 1 : 5. Find the largest side of the sheet.
Sides = x and 5x
Now vol. of rectangle = vol. of cube
x * 5x * (0.5) = 10*10 *10
Solve, x = 20
Largest side = 5x = 5*20
The area of the inner part of a cylinder is 616 sq. cms and its radius is half its height. Find the inner volume of the cylinder.
Given 2ᴨrh + ᴨr2 = 616 and r = (1/2) * h
So 2ᴨ × (1/2)h × h + ᴨ × (1/4)h2 = 616
Solve, h = 28/√5
Volume = ᴨr2h = (22/7) * (1/4) * h2 * h
Put h = 28/√5, vol. ≈ 1538.5
A cylinder and a cone have equal base and equal height. The ratio of the radius of base to height is 5 : 12. Find the ratio of the total surface area of the cylinder to that of the cone.
Let radius = 5x and height = 12x
Then slant height = √[(5x)2 + (12x)2]= 13x
Required ratio = 2ᴨr(h+r) : ᴨr(l+r)
A cone of radius 12 cm and height 5 cm is mounted on a cylinder of radius 12 cm and height 19 cm. Find the total surface area of the figure thus formed.
Slant height of cone, l = √(122 + 52) = 13 cm
Total surface area of final figure = curved surface area of cone + curved surface area of cylinder + area of base
= ᴨrl + 2ᴨrh + ᴨr2
= ᴨr (l + 2h + r)
= (22/7) * 12 (13 + 2*19 +12)
How many spherical balls whose radius is half that of cylinder can be formed by melting a cylindrical iron rod whose height is eight times its radius?
Let radius of rod = r, then height = 8r
Radius of 1 spherical ball = r/2
So number of balls = Vol. of cylindrical rod/Vol. of 1 spherical ball
= ᴨ × r2 × 8r / (4/3) × ᴨ × (r/2)3
A rectangle garden having length and breadth as 110 m and 65 m respectively has 2.5 m wide path around the sides inside the garden. Find the cost of gravelling the path at 50 paise per sq. metre.
Area of garden = 110*65 = 7150 m2
Area of path = (110-5)*(65-5) = 6300 m2
So area of path = 7150 – 6300 = 850 m2
So cost = 850*(50/100) = Rs 425
The cost of painting a triangular board at Rs 24.68 per m2 is Rs 33318. If the base of this board is thrice its height, find the height of the board.
Area of triangle = 33318/24.68 = 1350 m2
If height is x, then base = 3x
So (1/2)*3x*x = 1350
x2 = 900, x = 30
so base = 3*30 = 90
The radius of a wheel of car is 70 cm. How many revolutions per minute the wheel will make in order to keep a speed of 66 km/hr?
Distance to be covered in 1 min = 66*(1000/60) = 1100 m
70 cm = 0.70 m
Circumference of wheel = 2*(22/7)*0.70 = 4.4 m
Number of revolutions = (1100/4.4) = 250
The length and breadth of a rectangle are in the ratio 3 : 2. If the length is increased by 5 m keeping the breadth same, the new area of rectangle is 2600 m2. What is the breadth of the rectangle?
3x, 2x
So (3x+5)*2x = 2600
6x2 + 10x = 2600
3x2 + 5x – 1300 = 0
3x2 – 60x + 65x – 1300 = 0
3x(x-20) + 65(x-20) = 0
So breadth = 20 m
What will be the percentage increase in the surface area of the cube whose side is increased by 50%.
Surface area of cube = 6a2
A increases by 50%
So area increases by 50 + 50 + (50)(50)/100 = 125%
The radius of base and height of a cylinder are in the ratio 2 : 3. Find the total surface area of the cylinder if its volume is 12936 cm3.
(22/7)*(2x)2*3x = 12936
Solve, x = 7
So radius = 7 cm, height = 21 cm
Total surface area = 2*(22/7)*14*(21+14) = 3080
The diameter and the height of a right circular cylinder are 11.2 cm and 21 cm respectively. The metal around its outer body is 0.4 cm thick. What is the volume of the metal?
Internal radius = (11.2)/2 = 5.6, so outer radius = 5.6+0.4 = 6cm
So volume of metal = (22/7)*62*21 – (22/7)*(5.6)2*21 = (22/7)*[62 – (5.6)2]*21
What is the volume of a right circular cone whose radius of base is 70 cm and curved surface area is 40040 cm2?
r = 70
(22/7)*70*l = 40040
So l = 182
h = √(182) 2 – 702 =168 cm
so vol = (1/3)*(22/7)*702 *168
Find the radius of each of the 8 spherical balls which are made from a solid sphere of radius 10 cm by melting it.
Volume of each of the 8 ball = [(4/3)*(22/7)*103]/8
So (4/3)*(22/7)*r3 = [(4/3)*(22/7)*103]/8
Solve, r = 5
The metal used in the cylinder having external radius 6 cm, height 15 cm and thickness 0.25 cm is to be cast from a cylinder of radius 1 cm. What is the approximate height of the cylinder from which casting is to be done?
Volume of metal = (22/7)*[62 – (5.75)2]*15
So (22/7)*12*h = (22/7)*[62 – (5.75)2]*15
Solve, h ≈ 44
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