★ Maths Problem !!
1: Is it possible to design a rectangular park of perimeter 80 m and area 400 m ? If so, find its length and breadth.
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Answers
Answer:
Hi..
Step-by-step explanation:
Let the length be l m and the breadth be b m.
Then the area would be lb=400
Perimeter would be 2(l+b)=80
lb=400
⇒2(l+b)=80
⇒l+b=40
∴b=40−l --(1)
Substituting (1) in Area, we get
⇒l(40−l)=400
⇒40l−l
2
=400
⇒l
2
−40l+400=0
⇒(l−20)(l−20)=0
∴l=20
has equal roots, so it is possible to design the rectangle of given parameters.
⇒b=40−20=20
We now know that the length of the park is 20 m and the breadth of the park is also 20 m.
Topic :-
- Area & Perimeter
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Question :-
Is it possible to design a rectangular park of perimeter 80 m and area 400 m ? If so, find its length and breadth.
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Solution :-
Given Information :-
- Perimeter of rectangular park = 80 m
- Area of rectangular park = 400 m²
Calculation :-
⇒ Perimeter of rectangular park = 80 m
⇒ Perimeter of rectangle = 2 ( l + b )
⇒ 2 ( l + b ) = 80 m
Transposing 2 to R.H.S.
⇒ l + b = (80 / 2) m
⇒ l + b = 40 m
∴ l = 40 - b
Area of rectangle = l × b
Now, Substituting the value of 'l' = '40 - b' in are formula of area of rectangle.
⇒ Area = ( 40 - b ) × ( b ) m²
⇒ 400 m² = ( 40 - b ) × ( b ) m²
⇒ ( - b² + 40 b - 400 = 0 ) × ( - 1 )
⇒ b² - 40 b + 400 = 0
⇒ ( b )² - 2 ( b )( 20 ) + ( 20 )² = 0
[ ∵ a² - 2ab + b² = ( a - b )²]
⇒ ( b - 20 )² = 0
∴ b - 20 = 0
∴ b - 20 = 0 ⇒ b = 20 m
AND
∴ b - 20 = 0
∴ b - 20 = 0 ⇒ b = 20 m
Hence, Breadth of the park = b = 20 m
And Length of the park = ( 40 - b ) = 20 m
Since, Length and Breadth are equal, therefore the rectangular park is actually a square park.
Reason :- A square is also a special type of rectangle, having all 4 sides equal in length and equidistant from each other. Each line meet the other at 90°. So the angle between two adjacent sides of a square is 90°.
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Final Answer :-
- Yes it is possible to design a rectangular park having it's area as 400 m² and perimeter as 80 m.
- The Length of such park will be 20 m.
- The Breadth of such park will be 20 m.
- Also, since, Length and Breadth are equal, so the rectangle will be a square (special type of rectangle).