An envelope has an area of $50$ square inches and a perimeter of $30$ inches. What are the dimensions of the envelope? $\newline$ $\_\_\_$ inches by $\_\_\_$ inches

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Q. An envelope has an area of

50

square inches and a perimeter of

30

inches. What are the dimensions of the envelope?

\newline

\_\_\_

inches by

\_\_\_

inches

Define Variables: Let's denote the length of the envelope as $l$ and the width as $w$ . The area $A$ of a rectangle is given by the formula $A = l \times w$ , and the perimeter $P$ is given by $P = 2l + 2w$ .
Area and Perimeter Equations: We are given the area of the envelope as $50$ square inches, so we have the equation $l \times w = 50$ .
Solve System of Equations: We are also given the perimeter of the envelope as $30$ inches, so we have the equation $2l + 2w = 30$ . We can simplify this equation by dividing both sides by $2$ , which gives us $l + w = 15$ .
Quadratic Equation Simplification: Now we have a system of two equations: $\newline$ $1$ . $l \times w = 50$ $\newline$ $2$ . $l + w = 15$ $\newline$ We can solve this system by expressing one variable in terms of the other using the second equation. Let's solve for $w$ : $w = 15 - l$ .
Factor Quadratic Equation: Substitute $w$ from the second equation into the first equation: $\newline$ $l \times (15 - l) = 50$ $\newline$ This simplifies to a quadratic equation: $\newline$ $l^2 - 15l + 50 = 0$
Find Possible Solutions: To solve the quadratic equation, we can factor it: $\newline$ $(l - 10)(l - 5) = 0$ $\newline$ This gives us two possible solutions for $l$ : $l = 10$ or $l = 5$ .
Find Possible Solutions: To solve the quadratic equation, we can factor it: $\newline$ $(l - 10)(l - 5) = 0$ $\newline$ This gives us two possible solutions for $l$ : $l = 10$ or $l = 5$ .If $l = 10$ , then $w = 15 - l = 15 - 10 = 5$ . $\newline$ If $l = 5$ , then $w = 15 - l = 15 - 5 = 10$ . $\newline$ Both pairs ( $l = 10$ , $w = 5$ ) and ( $l = 5$ , $l$ $1$ ) are valid solutions because they are interchangeable due to the envelope's length and width being relative terms.