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A rectangular goat pen has an area of $24\,\text{square meters}$ and a perimeter of $20\,\text{meters}$ . What are the dimensions of the pen? $\newline$ $\_\_\_\_$ meters by $\_\_\_\_$ meters

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Area and perimeter: word problems

Full solution

Q. A rectangular goat pen has an area of

24\,\text{square meters}

and a perimeter of

20\,\text{meters}

. What are the dimensions of the pen?

\newline

\_\_\_\_

meters by

\_\_\_\_

meters

Define Area Formula: Let $l$ and $w$ be the length and width of the rectangular goat pen, respectively. The area of a rectangle is given by the formula $\text{Area} = l \times w$ .
Area Equation: Given the area is $24$ square meters, we have the equation $24 = l \times w$ .
Define Perimeter Formula: The perimeter of a rectangle is given by the formula $\text{Perimeter} = 2(l + w)$ . Given the perimeter is $20$ meters, we have the equation $20 = 2(l + w)$ .
Perimeter Equation: Simplify the perimeter equation to find $l + w$ . $20 = 2(l + w)$ simplifies to $10 = l + w$ .
Simplify Perimeter: We now have two equations: $24 = l \times w$ and $10 = l + w$ . We can solve these equations simultaneously. Substitute $w = 10 - l$ into the area equation.
Substitute for Area: Substituting gives $24 = l \times (10 - l)$ . Expanding this, we get $24 = 10l - l^2$ . Rearranging gives $l^2 - 10l + 24 = 0$ .
Expand and Rearrange: Factorize the quadratic equation: l - 6)(l - 4) = 0\. So, \$l = 6 or $l = 4$ . If $l = 6$ , then $w = 10 - 6 = 4$ . If $l = 4$ , then $w = 10 - 4 = 6$ .

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