is it possible to design a rectangular park of perimeter 80 m and area 400 m²? if so,find its length and breadth
Answers
Answer :
Yes, it is possible to design a rectangular park of perimeter 80 m and area 400 m².
Dimensions of the rectanglular park :-
- Length = 20 m
- Breadth = 20 m
Given :
• Perimeter of the rectangular park = 80 m
• Area of the rectangular park = 400 m²
To find :
• The length and breadth of the rectangular park if it is possible to design the rectangular park of perimeter 80 m and 400 m²
Concept :
Here, we have to find the length and breadth of the rectangular park. So, firstly we will assume the length of the rectangular park as x. And by using the formula of perimeter of the rectangle we will get the expression for the breadth. Then by using the formula of area of rectangle and substituting the given values we will get the value of 'x'. Substitute the value of x in the dimensions of the rectangular park which we've let. The resultant values will be the required answer.
• Formula to find perimeter of rectangle = 2(l + b)
• Formula to find area of rectangle = l × b
where,
• l denotes the length
• b denotes the breadth
Solution :
Let the length of the rectangular park be x metre.
→ Perimeter of rectangular park = 2(l + b)
→ Substituting the given values :-
→ 80 = 2(x + b)
→ Transposing 2 to the left hand side :-
→ 80 ÷ 2 = x + b
→ 40 = x + b
→ Transposing x to the left hand side and changing it's sign :-
→ 40 - x = b
The expression for the breadth of the rectangular park = 40 - x metre
→ Area of the rectanglular park = l × b
→ Substituting the given values :-
→ 400 = x * (40 - x)
→ 400 = 40x - x²
→ Transposing '40x - x²' to the left hand side :-
→ x² - 40x + 400 = 0
→ Now, quadratic equation is formed.
→ Product of the quadratic = 400x²
→ x² - 20x - 20x + 400 = 0
→ x(x - 20) - 20(x - 20) = 0
→ (x - 20)(x - 20) = 0
→ (x - 20) = 0 or (x - 20) = 0
→ x = 20 or x = 20
Therefore, the value of x = 20
Substitute the value of x in the dimensions of the rectanglular park :-
Length of the rectangular park :-
→ Length = x = 20 metre
Length of the rectanglular park = 20 metre
Breadth of the rectangular park :-
→ Breadth = 40 - x
→ Breadth = 40 - 20
→ Breadth = 20
Breadth of the rectanglular park = 20 metre
Therefore, the length and breadth of the rectangular park are 20 m and 20 m respectively.
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
VERIFICATION :-
We can verify the value of the length and breadth of the rectangular park by finding the area of the park. If it will be equal to 400 m² (as mentioned in the question) then the answer is right.
→ Area of the rectanglular park = l × b
→ Area of the rectanglular park = 20 × 20 = 400
Area of the rectangular park = 400 m²
Hence, verified.