Susan plans to use 120 feet of fencing to enclose a rectangular area for a garden. Which equation best models the area, $y$, of the rectangular garden that she creates if one side is $x$ feet long?
$y=(60-x)(x)$
$y=(120-x)(120+x)$
$y=(120-x)(x)$
$y=(60-x)(60+x)$
Answer (2)
Let the length of the rectangle be x and the width be w .
The perimeter of the rectangle is 2 x + 2 w = 120 .
Solve for w in terms of x : w = 60 − x .
The area of the rectangle is y = x c d o tw = x ( 60 − x ) . Therefore, y = ( 60 − x ) ( x ) .
The equation that models the area is y = ( 60 − x ) ( x )
Explanation
Problem Analysis Let's analyze the problem. Susan has 120 feet of fencing to enclose a rectangular garden. We need to find an equation that models the area, y , of the rectangular garden if one side is x feet long.
Perimeter Equation Let the length of the rectangle be x and the width be w . The perimeter of the rectangle is given by 2 x + 2 w = 120 . We can solve for w in terms of x .
Solving for Width Dividing the perimeter equation by 2, we get x + w = 60 . Solving for w , we have w = 60 − x .
Area Equation The area of the rectangle is given by y = x ⋅ w . Substituting w = 60 − x , we get y = x ( 60 − x ) , which can also be written as y = ( 60 − x ) ( x ) .
Final Equation Therefore, the equation that models the area of the rectangular garden is y = ( 60 − x ) ( x ) .
Examples
Imagine you're designing a garden and want to maximize the area you can enclose with a limited amount of fencing. This problem helps you understand how the dimensions of a rectangle affect its area, given a fixed perimeter. For example, if you have 120 feet of fencing, this equation helps you determine the best length and width to use for your garden to get the most planting space. This concept is also useful in other real-world scenarios, such as optimizing the layout of a room or designing a container with the largest possible volume using a limited amount of material.
The equation that models the area of Susan's garden is y = ( 60 − x ) ( x ) , where x represents one side of the garden. This outcome results from expressing the width in terms of the length due to the perimeter constraint of 120 feet. The selected option is y = ( 60 − x ) ( x ) .
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