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Anna has a 25 -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 25 square meters.
Write an equation in terms of
w that models the situation.

Anna has a $25$ -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 $25$ 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. Anna has a

25

-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

25

square meters.

\newline

Write an equation in terms of

w

that models the situation.

Perimeter of the garden: Let's denote the width of the garden as $w$ meters and the length as $l$ meters. The perimeter of the rectangle is the sum of all its sides, which is given by the formula $P = 2l + 2w$ . Since Anna has a $25$ -meter-long fence, the perimeter of the garden is $25$ meters. So we have: $\newline$ $P = 2l + 2w = 25$
Area of the garden: We also know that the area of the rectangle is given by the formula $A = l \times w$ . Anna wants the area of the garden to be $25$ square meters, so we have: $\newline$ $A = l \times w = 25$
Solving for l: Now we have two equations: $\newline$ $1$ ) $2l + 2w = 25$ (perimeter) $\newline$ $2$ ) $l \times w = 25$ (area) $\newline$ We can solve one of the equations for $l$ and then substitute it into the other equation. Let's solve the perimeter equation for $l$ : $\newline$ $2l = 25 - 2w$ $\newline$ $l = \frac{25 - 2w}{2}$
Substituting $l$ into the area equation: Substitute the expression for $l$ from the perimeter equation into the area equation: $\newline$ $l \cdot w = 25$ $\newline$ $\left(\frac{25 - 2w}{2}\right) \cdot w = 25$
Simplifying the equation: Now, we simplify the equation: $\newline$ $(25w - 2w^2) / 2 = 25$ $\newline$ Multiply both sides by $2$ to get rid of the fraction: $\newline$ $25w - 2w^2 = 50$
Rearranging the equation: Rearrange the equation to set it to zero: $\newline$ $-2w^2 + 25w - 50 = 0$
Correcting the error: This is a quadratic equation in terms of $w$ . However, we made a mistake in the previous step. The correct equation should be: $\newline$ $2w^2 - 25w + 50 = 0$ $\newline$ We need to correct this error.

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