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The perimeter of a chalkboard is $18$ feet. The area is $18$ square feet. What are the dimensions of the board? $\newline$ $\_\_\_$ feet by $\_\_\_$ feet

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Area and perimeter: word problems

Full solution

Q. The perimeter of a chalkboard is

18

feet. The area is

18

square feet. What are the dimensions of the board?

\newline

\_\_\_

feet by

\_\_\_

feet

Perimeter Equation: Let's denote the length of the chalkboard as $L$ and the width as $W$ . The perimeter ( $P$ ) of a rectangle is given by the formula $P = 2(L + W)$ . We are given that the perimeter is $18$ feet, so we have: $\newline$ $2(L + W) = 18$
Isolating Variables: To find the length and width, we need to divide both sides of the perimeter equation by $2$ to isolate the $L + W$ term: $\newline$ $L + W = \frac{18}{2}$ $\newline$ $L + W = 9$
Area Equation: The area $A$ of a rectangle is given by the formula $A = L \times W$ . We are given that the area is $18$ square feet, so we have: $\newline$ $L \times W = 18$
System of Equations: Now we have a system of two equations with two variables: $\newline$ $1$ ) $L + W = 9$ $\newline$ $2$ ) $L \times W = 18$ $\newline$ We can solve this system by substitution or elimination. Let's try substitution. If we solve the first equation for $W$ , we get: $\newline$ $W = 9 - L$
Substitution: Substitute $W = 9 - L$ into the second equation $L \times W = 18$ : $L \times (9 - L) = 18$ Expanding this, we get: $9L - L^2 = 18$
Quadratic Equation: Rearrange the equation to form a quadratic equation: $L^2 - 9L + 18 = 0$
Factoring: We can factor this quadratic equation: $\newline$ $(L - 3)(L - 6) = 0$
Solving for L: Setting each factor equal to zero gives us the possible values for L: $\newline$ $L - 3 = 0$ or $L - 6 = 0$ $\newline$ So, $L = 3$ or $L = 6$
Checking Solutions: If $L = 3$ , then $W = 9 - L = 9 - 3 = 6$ . If $L = 6$ , then $W = 9 - L = 9 - 6 = 3$ . Both pairs $(L = 3, W = 6)$ and $(L = 6, W = 3)$ satisfy the system of equations, so the dimensions of the chalkboard can be $3$ feet by $6$ feet or $6$ feet by $3$ feet.

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