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A rectangular field has area equal to 150 m2 and perimeter 50 m. its length and breadth respectively must be
- a)30 m, 5 m
- b)25 m, 6 m
- c)15 m, 10 m
- d)none of the above.
Correct answer is option 'C'. Can you explain this answer?
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A rectangular field has area equal to 150m2and perimeter 50 m. its len...
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Problem:
A rectangular field has an area equal to 150m2 and a perimeter of 50m. What are the length and breadth of the field?
Solution:
To solve this problem, we need to use the given information to set up a system of equations and solve for the length and breadth of the rectangular field.
Step 1: Set up the equations:
Let's assume the length of the field is 'l' and the breadth is 'b'.
We are given two pieces of information:
1. The area of the field is 150m2:
Area = length × breadth
150 = l × b
2. The perimeter of the field is 50m:
Perimeter = 2 × (length + breadth)
50 = 2 × (l + b)
Step 2: Solve the system of equations:
Now we have a system of two equations with two variables. We can solve this system to find the values of 'l' and 'b'.
Method 1: Substitution method
From the first equation, we can express 'l' in terms of 'b':
l = 150/b
Substituting this value of 'l' into the second equation:
50 = 2 × (150/b + b)
Simplifying the equation:
50 = 300/b + 2b
Multiplying through by 'b' to eliminate the fraction:
50b = 300 + 2b^2
Rearranging the equation:
2b^2 - 50b + 300 = 0
Solving this quadratic equation using factoring, the quadratic formula, or completing the square, we find that:
b = 10 or b = 15
If b = 10, then l = 150/10 = 15
If b = 15, then l = 150/15 = 10
Method 2: Elimination method
We can eliminate one variable by multiplying one equation by a suitable factor to make the coefficients of one variable in both equations the same.
From the first equation, we have:
150 = l × b [Equation 1]
From the second equation, we have:
50 = 2 × (l + b)
25 = l + b [Equation 2]
Now, we can multiply Equation 2 by 'l':
25l = l × l + l × b
Substituting the value of 'lb' from Equation 1 into the above equation:
25l = l^2 + 150
Rearranging the equation:
l^2 - 25l + 150 = 0
Factoring the quadratic equation:
(l - 15)(l - 10) = 0
So, l = 10 or l = 15
Step 3: Determine the length and breadth:
From the solutions obtained using both methods, we get two possible pairs of length and breadth values:
If b = 10, then l = 15
If b = 15, then l = 10
However, we need to choose the values that make sense in the context of a rectangular field. Since length is typically greater than the breadth, we can conclude that the length of the field is 15
A rectangular field has an area equal to 150m2 and a perimeter of 50m. What are the length and breadth of the field?
Solution:
To solve this problem, we need to use the given information to set up a system of equations and solve for the length and breadth of the rectangular field.
Step 1: Set up the equations:
Let's assume the length of the field is 'l' and the breadth is 'b'.
We are given two pieces of information:
1. The area of the field is 150m2:
Area = length × breadth
150 = l × b
2. The perimeter of the field is 50m:
Perimeter = 2 × (length + breadth)
50 = 2 × (l + b)
Step 2: Solve the system of equations:
Now we have a system of two equations with two variables. We can solve this system to find the values of 'l' and 'b'.
Method 1: Substitution method
From the first equation, we can express 'l' in terms of 'b':
l = 150/b
Substituting this value of 'l' into the second equation:
50 = 2 × (150/b + b)
Simplifying the equation:
50 = 300/b + 2b
Multiplying through by 'b' to eliminate the fraction:
50b = 300 + 2b^2
Rearranging the equation:
2b^2 - 50b + 300 = 0
Solving this quadratic equation using factoring, the quadratic formula, or completing the square, we find that:
b = 10 or b = 15
If b = 10, then l = 150/10 = 15
If b = 15, then l = 150/15 = 10
Method 2: Elimination method
We can eliminate one variable by multiplying one equation by a suitable factor to make the coefficients of one variable in both equations the same.
From the first equation, we have:
150 = l × b [Equation 1]
From the second equation, we have:
50 = 2 × (l + b)
25 = l + b [Equation 2]
Now, we can multiply Equation 2 by 'l':
25l = l × l + l × b
Substituting the value of 'lb' from Equation 1 into the above equation:
25l = l^2 + 150
Rearranging the equation:
l^2 - 25l + 150 = 0
Factoring the quadratic equation:
(l - 15)(l - 10) = 0
So, l = 10 or l = 15
Step 3: Determine the length and breadth:
From the solutions obtained using both methods, we get two possible pairs of length and breadth values:
If b = 10, then l = 15
If b = 15, then l = 10
However, we need to choose the values that make sense in the context of a rectangular field. Since length is typically greater than the breadth, we can conclude that the length of the field is 15
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A rectangular field has area equal to 150m2and perimeter 50 m. its length and breadth respectively must bea)30 m, 5 mb)25 m, 6 mc)15 m, 10 md)none of the above.Correct answer is option 'C'. Can you explain this answer?
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