A rectangular lawn in front of the Parliament House has a perimeter of 42 m and an area of [tex]$68 m^2$[/tex]. Find the length and width of the lawn.
Answer (2)
Define length l and width w of the rectangle.
Express the perimeter as 2 ( l + w ) = 42 and the area as lw = 68 .
Solve the system of equations to find l and w .
Determine the length and width: l = 17 m , w = 4 m .
Explanation
Problem Analysis Let's analyze the problem. We have a rectangular lawn with a given perimeter and area. We need to find the length and width of the lawn.
Formulate Equations Let l be the length and w be the width of the rectangular lawn. The perimeter is given by 2 ( l + w ) = 42 which simplifies to l + w = 21 The area is given by lw = 68
Express width in terms of length We have a system of two equations with two variables:
l + w = 21
lw = 68
From the first equation, we can express w in terms of l :
w = 21 − l
Formulate Quadratic Equation Substitute this expression for w into the second equation: l ( 21 − l ) = 68 This results in a quadratic equation: 21 l − l 2 = 68 Rearrange the equation to the standard quadratic form: l 2 − 21 l + 68 = 0
Solve Quadratic Equation Now, we solve the quadratic equation l 2 − 21 l + 68 = 0 . We can factor this equation as: ( l − 4 ) ( l − 17 ) = 0 Thus, the solutions for l are: l = 4 or l = 17
Find the width If l = 4 , then w = 21 − l = 21 − 4 = 17 If l = 17 , then w = 21 − l = 21 − 17 = 4 In either case, the length and width are 17 m and 4 m.
Final Answer Therefore, the length and width of the rectangular lawn are 17 m and 4 m (or vice versa).
Examples
Understanding how to calculate the dimensions of a rectangle given its perimeter and area is useful in many real-world scenarios. For example, when planning a garden, you might know how much fencing you have (the perimeter) and how much space you want to cover (the area). Using these calculations, you can determine the optimal dimensions of your garden to maximize space and resources. This ensures you can fit all your desired plants while staying within your fencing constraints.
The length and width of the rectangular lawn are 17 meters and 4 meters, or vice versa. This can be determined using the perimeter and area formulas to create a system of equations. Solving these equations provides the necessary dimensions.
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