T. $7$ Area and perimeter: word problems MHV $\newline$ A rectangular piece of metal has an area of $693$ square centimeters. Its perimeter is $172$ centimeters. What are the dimensions of the piece? $\newline$ $\square$ centimeters by $\square$ centimeters $\newline$ Submit

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Area and perimeter: word problems

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Q. T.

7

Area and perimeter: word problems MHV

\newline

A rectangular piece of metal has an area of

693

square centimeters. Its perimeter is

172

centimeters. What are the dimensions of the piece?

\newline

\square

centimeters by

\square

centimeters

\newline

Submit

Define Variables: Let's call the length of the rectangle $l$ and the width $w$ . The area of a rectangle is given by $\text{Area} = l \times w$ .
Area and Perimeter Equations: We know the area is $693$ square centimeters, so $l \times w = 693$ .
Simplify Perimeter Equation: The perimeter of a rectangle is given by Perimeter = $2l + 2w$ . We know the perimeter is $172$ centimeters, so $2l + 2w = 172$ .
System of Equations: Let's simplify the perimeter equation to $l + w = 86$ by dividing everything by $2$ .
Solve for Width: Now we have a system of two equations: $l \times w = 693$ and $l + w = 86$ .
Substitute and Expand: We can solve for one variable in terms of the other using the second equation. Let's solve for $w$ : $w = 86 - l$ .
Quadratic Equation: Substitute $w$ from $w = 86 - l$ into the area equation: $l \times (86 - l) = 693$ .
Factorization: Expand the equation: $86l - l^2 = 693$ .
Solve for Length: Rearrange the equation to form a quadratic equation: $l^2 - 86l + 693 = 0$ .
Check Dimensions: We can solve this quadratic equation by factoring or using the quadratic formula. Let's try factoring first.
Check Dimensions: We can solve this quadratic equation by factoring or using the quadratic formula. Let's try factoring first.The factors of $693$ that add up to $86$ are $21$ and $63$ . So the factored form is $(l - 21)(l - 63) = 0$ .
Check Dimensions: We can solve this quadratic equation by factoring or using the quadratic formula. Let's try factoring first.The factors of $693$ that add up to $86$ are $21$ and $63$ . So the factored form is $(l - 21)(l - 63) = 0$ .Set each factor equal to zero and solve for $l$ : $l - 21 = 0$ or $l - 63 = 0$ . This gives us $l = 21$ or $l = 63$ .
Check Dimensions: We can solve this quadratic equation by factoring or using the quadratic formula. Let's try factoring first.The factors of $693$ that add up to $86$ are $21$ and $63$ . So the factored form is $(l - 21)(l - 63) = 0$ .Set each factor equal to zero and solve for $l$ : $l - 21 = 0$ or $l - 63 = 0$ . This gives us $l = 21$ or $l = 63$ .If $l = 21$ , then $86$ $1$ . If $l = 63$ , then $86$ $3$ .
Check Dimensions: We can solve this quadratic equation by factoring or using the quadratic formula. Let's try factoring first.The factors of $693$ that add up to $86$ are $21$ and $63$ . So the factored form is $(l - 21)(l - 63) = 0$ .Set each factor equal to zero and solve for $l$ : $l - 21 = 0$ or $l - 63 = 0$ . This gives us $l = 21$ or $l = 63$ .If $l = 21$ , then $86$ $1$ . If $l = 63$ , then $86$ $3$ .We need to check both sets of dimensions $86$ $4$ and $86$ $5$ to see which one gives us the correct area and perimeter.
Check Dimensions: We can solve this quadratic equation by factoring or using the quadratic formula. Let's try factoring first.The factors of $693$ that add up to $86$ are $21$ and $63$ . So the factored form is $(l - 21)(l - 63) = 0$ .Set each factor equal to zero and solve for $l$ : $l - 21 = 0$ or $l - 63 = 0$ . This gives us $l = 21$ or $l = 63$ .If $l = 21$ , then $86$ $1$ . If $l = 63$ , then $86$ $3$ .We need to check both sets of dimensions $86$ $4$ and $86$ $5$ to see which one gives us the correct area and perimeter.For $86$ $4$ : Area check: $86$ $7$ which is not equal to $693$ . Perimeter check: $86$ $9$ which is correct. But since the area is incorrect, this set of dimensions is wrong.