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Olivia has a 20 -meter-long fence that she plans to use to enclose a rectangular garden of width
w. The fencing will be placed around all four sides of the garden so that its area is 18.75 square meters.
Write an equation in terms of
w that models the situation.

Olivia has a $20$ -meter-long fence that she plans to use to enclose a rectangular garden of width $w$ . The fencing will be placed around all four sides of the garden so that its area is $18$ . $75$ square meters. $\newline$ Write an equation in terms of $w$ that models the situation.

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Write and solve equations for proportional relationships

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Q. Olivia has a

20

-meter-long fence that she plans to use to enclose a rectangular garden of width

w

. The fencing will be placed around all four sides of the garden so that its area is

18

.

75

square meters.

\newline

Write an equation in terms of

w

that models the situation.

Perimeter equation: Let's denote the width of the garden as $w$ meters and the length as $l$ meters. The perimeter of the garden is the sum of all four sides, which is given as $20$ meters. The perimeter $P$ of a rectangle is given by $P = 2l + 2w$ . Since we know the total perimeter is $20$ meters, we can write the equation: $\newline$ $2l + 2w = 20$
Simplifying the equation: We can simplify this equation by dividing everything by $2$ to make it easier to solve for one of the variables: $\newline$ $l + w = 10$ $\newline$ Now we have an equation relating the length and width of the garden.
Area equation: We also know the area $A$ of the garden is given by $A = l \times w$ . The problem states that the area is $18.75$ square meters, so we can write the equation: $\newline$ $l \times w = 18.75$
Expressing $l$ in terms of $w$ : We can use the perimeter equation to express $l$ in terms of $w$ . From the simplified perimeter equation $l + w = 10$ , we can solve for $l$ : $\newline$ $l = 10 - w$
Equation in terms of $w$ : Now we can substitute the expression for $l$ into the area equation to write an equation only in terms of $w$ : $\newline$ $(10 - w) \cdot w = 18.75$ $\newline$ This is the equation that models the situation in terms of $w$ .

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