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The area of a rectangular restaurant kitchen is $70\,\text{square meters}$ . The perimeter is $34\,\text{meters}$ . What are the dimensions of the kitchen? $\newline$ $\_\_\_\_$ meters by $\_\_\_\_$ meters

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Area and perimeter: word problems

Full solution

Q. The area of a rectangular restaurant kitchen is

70\,\text{square meters}

. The perimeter is

34\,\text{meters}

. What are the dimensions of the kitchen?

\newline

\_\_\_\_

meters by

\_\_\_\_

meters

Rectangle Area Formula: Let $l$ and $w$ be the length and width of the rectangular kitchen, respectively. $\newline$ We know the area $A$ of a rectangle is given by $A = l \times w$ . $\newline$ Given area: $70$ square meters. $\newline$ $70 = l \times w$
Rectangle Perimeter Formula: The perimeter $P$ of a rectangle is given by $P = 2l + 2w$ . $\newline$ Given perimeter: $34$ meters. $\newline$ $34 = 2l + 2w$
Simplify Perimeter Equation: We can simplify the perimeter equation to make one variable the subject. $\newline$ Divide the perimeter equation by $2$ : $\newline$ $17 = l + w$ $\newline$ Now, substitute $w = 17 - l$ into the area equation: $\newline$ $70 = l \times (17 - l)$
Solve Quadratic Equation: Solve the quadratic equation $70 = l \times (17 - l)$ . $\newline$ Expanding it gives: $\newline$ $70 = 17l - l^2$ $\newline$ Rearrange to form a standard quadratic equation: $\newline$ $l^2 - 17l + 70 = 0$
Factorize Quadratic Equation: Factorize the quadratic equation: $\newline$ $(l - 7)(l - 10) = 0$ $\newline$ So, $l = 7$ or $l = 10$ . $\newline$ If $l = 7$ , then $w = 17 - 7 = 10$ . $\newline$ If $l = 10$ , then $w = 17 - 10 = 7$ . $\newline$ Both pairs $(7, 10)$ and $(10, 7)$ are valid as dimensions of the rectangle.

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