WORD PROBLEMS
RELATED TO MENSURATION
CLASS 8th, 9th AND 10th
Q.1) Find the total surface area and the length of the diagonal of a cube of side 27cm.
SOLUTION:
Given:
Side of the cube = 27cm
Total surface area of the cube = 6 * Side2
= 6 * (27cm)2 = 6 * 729 cm2 = 4,374 cm2
Length of diagonal = √3 * Side = √3 * 27cm = 27√3 cm
Hence, the total surface area of the cube is 4,374 cm2 and its diagonal is 27√3 cm long.
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Q.2) Find the length of the longest pole that can be put in a room of dimensions 8m by 8m by 4m.
SOLUTION:
Given:
Length = L = 8m
Breadth = B = 8m
Height = H = 4m
Length of the longest pole = Length of the diagonal room
= √L2 + √B2 + √H2
= √(8m)2 + √(8m)2 + √(4m)2
= √64 + √64 + √16m
= √144m = 12m
Hence, the required length of the longest pole is 12m.
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Q.3) The dimensions of a cuboid are in the ratio 4 : 3 : 2. If the total surface area of the cuboid is 208m2, find its dimensions.
SOLUTION:
Given:
Dimensions of a cuboid are in the ratio 4 : 3 : 2
Let the length = L = 4x
Let the breadth = B = 3x
Let the height = H = 2x
Then total surface area = 208
2 ( LB + BH + HL ) = 208
2 ( 4x * 3x + 3x * 2x + 2x * 4x ) = 208
2 ( 12x2 + 6x2 + 8x2 ) = 208
2 * 26x2 = 208
52x2 = 208
x2 = 208 / 52 = 4
x = 2
Therefore,
Length = 4x = 4 * x = 4 * 2 = 8m
Breadth = 3x = 3 *x = 3 * 2 = 6m
Height = 2x = 2 * x = 2 * 2 = 4m
Hence, the dimensions of the cuboid are 8m, 6m and 4m.
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Q.4) The internal measures of a cuboidal room are 12m * 8m * 4m. Find the total cost of whitewashing all four walls of the room if the cost of whitewashing is Rs. 5 per m2. What will be the cost of whitewashing if the ceiling of the room is also whitewashed ?
SOLUTION:
Given:
Length = L = 12m
Breadth = B = 8m
Height = H = 4m
Area of four walls of the room = Perimeter of the base * Height of the room
= 2 ( L + B ) * H
= 2 ( 12 + 8 ) * 4
= ( 2 * 20 * 4 ) m2
= 160 m2
Cost of whitewashing = Rs. 5 per m2 [ Given ]
The cost of whitewashing four walls of the room = Rs. ( 160 * 5 ) = Rs. 800
Area of ceiling = 12m * 8m = 96 m2
Cost of whitewashing the ceiling = Rs. ( 96 * 5 ) = Rs. 480
Hence, the total cost of whitewashing the room = Rs. ( 800 + 480 ) = Rs. 1,280
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Q.5) A room of dimensions 12m by 6m by 3m has two doors each of dimensions 2.5m * 1.5m and four windows each of dimensions 2m * 1.5m. What will be the cost of whitewashing the walls and roof at the rate of Rs. 10 per square meter ?
SOLUTION:
Given:
Length = L = 12m
Breadth = B = 6m
Height = H = 3m
Area of the four walls = 2H ( L + B )
= 2 * 3 ( 12 + 6 )
= 6m * 18m
= 108 m2
Area of the roof = L * B = 12 * 6 = 72 m2
Area of doors = 2 ( 2.5 * 1.5 ) = 2 * 3.75 = 7.5 m2
Area of windows = 4 ( 2 * 1.5 ) = 4 * 3 m2 = 12 m2
Area of the room to be whitewashed
= Area of four walls + Area of roof – ( Area of doors + Area of windows )
= 108 m2 + 72 m2 – ( 7.5 m2 + 12m2 )
= 180 m2 – 19.5 m2
= 160.50 m2
Cost of whitewashing = Rs. ( 160.50 * 10 ) = Rs. 1,605
Hence, the cost of whitewashing the room is Rs. 1,605.
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