Mensuration - Exercise 23(D)

A room is 10 m long and 6 m broad. It is surrounded by a verandah which is 2 m wide all around it. Find the cost of flooring the verandah with marble at ₹ 236 per m².

Answer

To find the area of the verandah, we calculate the area of the outer rectangle (room + verandah) and subtract the area of the inner rectangle (room).

Given:

Inner (Room): Length = 10 m, Breadth = 6 m

Outer (Room + Verandah): Since the verandah is 2 m wide on all sides, we add 2 + 2 = 4 m to both dimensions.

Outer Length = 10 + 4 = 14 m

Outer Breadth = 6 + 4 = 10 m

Rate of cost of flooring = ₹ 236 per m²

Area of Inner Room = Inner Length x Inner Breadth

= 10 x 6

= 60 m²

Area of Outer Rectangle = Inner Length x Inner Breadth

= 14 m x 10 m

= 140 m²

Area of Verandah = Outer Area - Inner Area

= 140 m² - 60 m²

= 80 m²

Cost of flooring the verandah:

Total cost = Area of Verandah x Rate

= ₹ 80 x 236

= ₹ 18880

The cost of flooring the verandah = ₹ 18880

A hall is 16 m long and 12 m broad. Find the cost of carpeting it at ₹ 615 per m², after leaving a margin of 1 metre all around.

Answer

In this case, the carpet is smaller than the room because a margin is left all around. We find the area of the carpeted portion.

Given:

Hall: Length = 16 m, Breadth = 12 m

Carpeted Area: A margin of 1 m is left on all sides, so we subtract 1 + 1 = 2 m from the hall's dimensions.

Carpet Length = 16 - 2 = 14 m

Carpet Breadth = 12 - 2 = 10 m

Rate of cost of carpeting = ₹ 615 per m²

Area of Carpet = Carpet Length x Carpet Breadth

= 14 m x 10 m

= 140 m²

Cost of carpeting:

Total cost = Area of Carpet x Rate

= ₹ 140 x 615

= ₹ 86100

The cost of carpeting = ₹ 86100

A sheet of paper measures 35 cm by 25 cm. A strip of 5 cm width is cut from it, all around. Find the area of the remaining sheet and also the area of the cut-out strip.

Answer

In this problem, the strip is removed from the inside of the original sheet.

Given:

Original Sheet: Length = 35 cm, Breadth = 25 cm

Remaining Sheet: A 5 cm strip is removed from all sides, so we subtract 5 + 5 = 10 cm from both dimensions.

Remaining Sheet Length = 35 - 10 = 25 cm

Remaining Sheet Breadth = 25 - 10 = 15 cm

Area of Remaining Sheet = Remaining Sheet Length x Remaining Sheet Breadth

= 25 cm x 15 cm

= 375 cm²

Area of Original Sheet = Original Sheet Length x Original Sheet Breadth

= 35 cm x 25 cm

= 875 cm²

Area of Cut-out Strip = Area of Original Sheet - Area of Remaining Sheet

= 875 cm² - 375 cm²

= 500 cm²

Area of Remaining Sheet = 375 cm², Area of Cut-out Strip = 500 cm²

A path 3 m wide is running along the inside of the boundary of a rectangular field 116 m by 76 m. How much money is needed to gravel the path at ₹ 42.50 per m²?

Answer

Given:

Outer Field: Length = 116 m, Breadth = 76 m

Inner Field (excluding path): Subtract 2 x 3 m = 6 m from both dimensions.

Inner Length = 116 - 6 = 110 m

Inner Breadth = 76 - 6 = 70 m

Rate = ₹ 42.50 per m²

Area of Outer Field = Outer Length x Outer Breadth

= 116 m x 76 m

= 8816 m²

Area of Inner Field = Inner Length x Inner Breadth

= 110 m x 70 m = 7700 m²

Area of Path = Outer Area - Inner Area

= 8816 m² - 7700 m²

= 1116 m²

Cost needed to gravel the path:

Total cost = Area of Path x Rate

= ₹ 1116 x 42.50

= ₹ 47430

Money needed to gravel the path = ₹ 47430

A path 2.5 m wide is running around a rectangular grassy plot 40 m by 35 m. Find the area of the path and the money needed for tilling it at ₹ 115.60 per m².

Answer

Given:

Inner Plot: Length = 35 m, Breadth = 40 m

Outer Rectangle (Plot + Path): Add 2 x 2.5 m = 5 m to both dimensions.

Outer Length = 35 + 5 = 40 m

Outer Breadth = 40 + 5 = 45 m

Rate = ₹ 115.60 per m²

Area of Outer Rectangle = Outer Length x Outer Breadth

= 40 m x 45 m = 1800 m²

Area of Inner Plot = Inner Length x Inner Length

= 35 m x 40 m = 1400 m²

Area of Path = Area of Outer Rectangle - Area of Inner Plot

= 1800 m² - 1400 m²

= 400 m²

Money needed for tilling the path = Area of Path x Rate

= ₹ 400 x 115.60

= ₹ 46240

Area of Path = 400 m², Money needed for tilling the path = ₹ 46240

A rectangular lawn 115 m long and 64 m broad has two cross-paths at right-angles, one 2 m wide, running parallel to its length and the other 2.5 m wide, running parallel to its breadth. Find the cost of gravelling the paths at ₹ 114 per m².

Answer

Let ABCD be one path:

Length = 115 m

Breadth = 2 m

Area of ABCD = Length x Breadth

= 115 m x 2 m

= 230 m²

Let EFGH be another path:

Length = 64 m

Breadth = 2.5 m

Area of EFGH = Length x Breadth

= 64 m x 2.5 m

= 160 m²

Then, JKLM is the area common to both:

Length = 2.5 m

Breadth = 2 m

Area of JKLM = Length x Breadth

= 2.5 m x 2 m

= 5 m²

∴ Area of cross-paths = Area ABCD + Area EFGH - Area JKLM

= 230 m² + 160 m² - 5 m²

= 390 m² - 5 m²

= 385 m²

Cost of gravelling the paths:

Rate of gravelling = ₹ 114 per m²

Total cost = Area of cross-paths x Rate

= ₹ 385 x 114

= ₹ 43890

The cost of gravelling the paths = ₹ 43890

The central hall of a school is 22 m long and 15.5 m wide. A carpet is to be laid on the floor leaving a strip of 75 cm width from the walls uncovered. Find the area of the carpet and the area of the strip left uncovered.

Answer

Given:

Length of the hall = 22 m

Breadth of the hall = 15.5 m

Area of the hall = Length x Breadth

= 22 m x 15.5 m

= 341 m²

The carpet leaves a strip of 75 cm uncovered all around.

Converting strip width to m:

1 m = 100 cm

∴ 75 cm = 0.75 m

Since the strip is on all sides, we must subtract the width twice from both the length and the breadth of the hall:

Length of the carpet = 22 m - (0.75 m + 0.75 m)

= 22 m - 1.5 m

= 20.5 m

Breadth of the carpet = 15.5 m - (0.75 m + 0.75 m)

= 15.5 m - 1.5 m

= 14 m

Area of the carpet = Length of the carpet x Breadth of the carpet

= 20.5 m x 14 m

= 287 m²

Area of the Uncovered Strip: The strip is the area between the hall floor and the carpet.

Area of strip = Area of hall floor - Area of carpet

= 341 m² - 287 m²

= 54 m²

Area of the carpet = 287 m², Area of strip = 54 m²

Find the area of the shaded region in each of the following figures :

(i)

(ii)

Answer

(i)

From the figure,

Inner Rectangle Dimensions:

Inner Length = 12 m

Inner Breadth = 9 m

Area of Inner rectangle = Inner Length x Inner Breadth

= 12 m x 9 m

= 108 m²

Outer Rectangle Dimensions:

To find the outer dimensions, add the width of the shaded border to both sides:

Outer Length = 12 m + (3 m + 3 m )

= 12 m + 6 m

= 18 m

Outer Breadth = 9 m + (2 m + 2 m)

= 9 m + 4 m

= 13 m

Area of Outer rectangle = Outer Length x Outer Breadth

= 18 m x 13 m

= 234 m²

Area of shaded region = Area of Outer rectangle - Area of Inner rectangle

= 234 m² - 108 m²

= 126 m²

Area of shaded region = 126 m²

(ii)

From the figure,

Horizontal path dimensions:

Length = 150 m

Breadth = 2.5 m

Area 1 = Length x Breadth

= 150 m x 2.5 m

= 375 m²

Vertical path dimensions:

Length = 50 m

Breadth = 1.6 m

Area 2 = Length x Breadth

= 50 m x 1.6 m

= 80 m²

Overlapping area: The overlap is a rectangle where the widths of the two paths meet.

Overlapping area = 2.5 m x 1.6 m

= 4 m²

Area of the shaded region = (Area 1 + Area 2) - Overlapping area

= (375 m² + 80 m²) - 4 m²

= 455 m² - 4 m²

= 451 m²

Area of shaded region = 451 m²