solution of NCERT Exercise 11.4
Perimeter and Area 7th Math
solution of NCERT Exercise 11.4
Question (1) A garden is 90 m long and 75 m broad. A path 5 m wide is to be built outside and around it. Find the area of the path. Also find the area of the garden in hectare.
Solution:
Let ABCD is the given rectangular garden.
In which
Length, AB = DC = 90 m
And, width, BC = AD = 75 m
A 5 m path is built outside and around the given garden.
Thus, width of the path = 5 m
Now, Length of the park along the path = 90 m + 5m +5m = 100 m
And, width of the park along the path = 75 m + 5m +5m = 85 m
Thus, area of path in hectare = ?
We know that, Area of rectangle = Length × Breadth
Thus, area of given rectangular park = 90 m × 75 m
⇒ Area of garden = 6750 m2
Now, Area of garden along with path = Length × Breadth
= 100 m × 85 m
⇒ Area of garden with path = 8500 m
Now, Area of path = Area of garden with path – Area of garden
= 8500 m – 6750 m
Thus, Area of path = 1750 m2
Area of path in Hectare
∵ We know that, 10000 m2 = 1 hectare
∴ 1 m2 = `1/10000` hectare
∴ 1750 m2 = `1/10000xx1750` hectare = 0.175 hectare
Thus, Area of path = 0.175 hectare Answer
Question (2) A 3 m wide path runs outside and around a rectangular park of length 125 m and breadth 65 m. Find the area of the path.
Solution :
Let ABCD is the given rectangular park.
In which
Length, AB = DC = 125 m
And, width, BC = AD = 65 m
A 3 m path is built outside and around the given park.
Thus, width of the path = 3 m
Now, Length of the park along the path = 125 m + 3m + 3m = 131 m
And, width of the park along the path = 65 m + 3m + 3m = 71 m
Thus, area of path = ?
We know that, Area of rectangle = Length × Breadth
Thus, area of given rectangular park = 125 m × 65 m
⇒ Area of park = 8125 m2
Now, Area of park along with path = Length × Breadth
= 131 m × 71 m
⇒ Area of park with path = 9301 m
Now, Area of path = Area of park with path – Area of park
= 9301 m – 8125 m
Thus, Area of path = 1176 m2 Answer
Question (3) A picture is painted on a cardboard 8 cm long and 5 cm wide such that there is a margin of 1.5 cm along each of its sides. Find the total area of the margin.
Solution :
Let EFGH is the cardboard, in which
Given,
Length (HG) = 8 cm
And width (EH) = 5 cm
And margin on cardboard = 1.5 cm
Thus, Total area of margin = ?
We know that,
Area of rectangle = Length × Width
Thus, Area of given cardboard = 8 cm × 5 cm
= 40 cm2
Now, Length picture painted on the cardboard, means length of cardboard without margin
= 8 cm – 1.5 cm – 1.5 cm
= 5 cm
And, width of the cardboard without margin
= 5 cm – 1.5 cm – 1.5 cm
= 5 cm – 3 cm
= 2 cm
Thus, Area of cardboard without margin = Length × Width
= 5 cm × 2 cm
= 10 cm
Thus area of cardboard without margin = 10 cm
Now, Area of margin = Area of cardboard – Area of cardboard without margin
= 40 cm – 10 cm
= 30 cm2
Thus, Area of margin = 30 cm2 Answer
Question (4) A verandah of width 2.25 m is constructed all along outside a room which is 5.5 m long and 4 m wide. Find:
(i) the area of the verandah
(ii) the cost of cementing the floor of the verandah at the rate of ₹ 200 per m2.
Solution:
Given, Length of the room = 5.5 m
Breadth of the room = 4 m
Width of verandah along outside of the room = 2.25 m
Rate of cementing the verandah = ₹ 200 per m2
Thus, Area of verandah and cost of cementing of verandah = ?
(i) Calculation of the area of the verandah
We know that, Area of rectangle = Length × Breadth
Thus, area of given room = 5.5 m × 4 m
= 22 m2
Now, length of the room including verandah
= 5.5 m + 2.25 m = 7.75 m
And, width of the room including verandah
= 4 m + 2.25 m = 6.25 m
Now, Area of room including verandah = length × width
= 7.75 m × 6.25 m
= 48.437 m2
Now, Area of verandah = Area of room including verandah – Area of room
= 48.437 m2 – 22 m2
= 26.437 m2
Thus, Area of verandah = 26.437 m2
(ii) Calculation of cost of cementing of verandah
∵ Cost of 1 m2 = ₹ 200
∴ cost of 26.437 m2 = ₹ 200 × 26.437 m2
= ₹ 5287.40
Thus, cost of cementing of verandah = ₹ 5287.40
Thus, Area of verandah =26.437 m2. And cost of cementing of verandah = ₹ 5287.40 Answer
Question (5) A path 1 m wide is build along the border and inside a square garden of side 30 m. Find
(i) the area of the path
(ii) the cost of planting grass in the remaining portion of the garden at the rate of ₹ 40 per m2.
Solution:
Given, side of square garden = 30 m
And width of path inside the garden = 1 m
Thus, side of garden without path = 28 m
Rate of plantation of garden = ₹ 40 per m2
Thus,
Area of path and cost of plantation of grass in the remaining portion of garden =?
(i) Area of path
We know that, Area of a square = (side) 2
Thus, area of given garden = (30 m )2
= 900 m2
Now, Area of garden without path = (28 m)2
= 784 m2
Thus, Area of path = Area of garden – Area of garden without path
= 900 m2 – 784 m2
= 116 m2
Thus, Area of path = 116 m2
(ii) Calculation of cost of plantation in the remaining garden
∵ cost of plantation for 1 square meter = ₹ 40
∴ cost of plantation for 784 m2 = ₹ 40 × 784 m2
= ₹ 31360.00
Thus, Area of path = 116 m2 And cost of plantation of remaining of garden = ₹ 31360.00 Answer
Question (6) Two cross roads, each of width 10 m, cut at right angles through the centre of a rectangular park of length 700 m and breadth 300 m and parallel to its sides. Find the area of the roads. Also find the area of the park excluding cross roads. Give the answer in hectares.
Solution :
Let ABCD is the rectangular park.
And road PS and road EF crosses each other at right angle.
In this according to question
Length of AB = 700 m
Length of BC = 300 m
And width of the road = 10 m
Since, roads crosses each other at right angle, thus they form a square KLMN in the middle
Since width of the road is equal to 10 m,
Thus side of the square = 10 m
Thus, Area of roads and area of park excluding roads = ?
We know that,
Area of rectangle = Length × Breadth
Thus, Area of given park = 700 m × 300 m
= 210000 m2
Calculation of area of road
Length of road parallel to length of the park = 700 m
And width of the road = 10 m
Thus, Area of road (EFGH) parallel the length of the park = 700 m × 10 m
= 7000 m2
Length of road parallel to width of the park = 300 m
Width of the road = 10 m
Thus, area of road (PQRS) parallel to width of the park = 300 m × 10 m
= 3000 m2
Calculation of area of square crosses in the middle of the road
Since, width of the road = 10 m,
Thus, side of the square (KLMN) = 10 m
We know that, area of square = side × side
Thus, area of square (KLMN) = 10 m × 10 m
= 100 m2
[Since two roads crosses each other, thus while finding the area of road, area of overlapping square is to be subtracted from total area of road.]
Now, total area of road
= area of road parallel to length (EFGH) + area of road parallel to width (PQRS) – Area of square (KLMN)
= 7000 m2 + 3000 m2 – 100 m2
= 100000 m2 – 100 m2
= 9900 m2
Thus, Area of roads = 9900 m2
Now, since, 10000 m2 = 1 hectare
Thus, 9900 m2 = `9900/10000` hectare
= 0.99 hectare
Thus, area of roads = 0.99 hectare
Now, Area of park excluding roads
= Area of park – Area of roads
= 210000 m2 – 9900 m2
= 200100 m2
Thus, area of park excluding roads = 200100 m2
Now, since 1 hectare = 10000 m2
∴ 200100m2 = 200100 × `1/10000` hectare
= 20.01 hectare
Thus, area of park excluding roads = 20.01 hectare and area of roads = 0.99 hectare Answer
Alternate Method
Let assume that roads are running just along the length and width.
Therefore, Length of the park without road
= 700 m –10 m = 690 m
And, breadth of the park without road
= 300 m – 10 m = 290 m
Now, area of the park = Length × Breadth
= 700 m × 300 m
= 210000 m2
And Area of park without road = Length × Breadth
= 690 m × 290 m
= 200100 m2
Now, since 1 hectare = 10000 m2
∴ 200100m2 = 200100 × `1/10000` hectare
= 20.01 hectare
Now, area of road = Area of park – Area of park without road
= 210000 – 200100 m2
= 900 m2
Thus, area of road = 900m2
Now, since, 10000 m2 = 1 hectare
Thus, 9900 m2 = `9900/10000` hectare
= 0.99 hectare
Thus, area of roads = 0.99 hectare
Thus, area of park excluding roads = 20.01 hectare and area of roads = 0.99 hectare Answer
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