The area of a rectangular garage is 30 square meters. The perimeter is 22 meters. What are the dimensions of the garage? _____ meters by _____ meters

Question

The area of a rectangular garage is 30 square meters.  The perimeter is 22 meters. What are the dimensions of the garage?  _____ meters by _____ meters

Bytelearn · Accepted Answer

Define Variables: Let l be the length and w be the width of the rectangular garage. The area of a rectangle is given by the formula A = l * w.Area and Perimeter Formulas: The area of the garage is given as 30 square meters. Substitute 30 for A in the formula A = l * w. 30 = l * wSubstitute Values: The perimeter of a rectangle is given by the formula P = 2l + 2w. The perimeter of the garage is given as 22 meters. Substitute 22 for P in the formula P = 2l + 2w. 22 = 2l + 2wSimplify Perimeter Equation: Divide the perimeter equation by 2 to simplify. 22 / 2 = (2l + 2w) / 2 11 = l + wSolve System of Equations: We now have two equations with two variables: 30 = l * w (Area equation) 11 = l + w (Perimeter equation) We can solve this system of equations to find the values of l and w.Express Width in Terms of Length: From the perimeter equation, express w in terms of l. w = 11 - lSubstitute Width into Area Equation: Substitute w = 11 - l into the area equation. 30 = l * (11 - l)Distribute and Rearrange Equation: Distribute l across the parentheses. 30 = 11l - l^2Factor Quadratic Equation: Rearrange the equation into a standard quadratic form. 0 = l^2 - 11l + 30Solve for Length: Factor the quadratic equation. (l - 5)(l - 6) = 0Find Width: Solve for l by setting each factor equal to zero. l - 5 = 0 or l - 6 = 0 l = 5 or l = 6Final Valid Solutions: If l = 5, then w = 11 - l = 11 - 5 = 6. If l = 6, then w = 11 - l = 11 - 6 = 5. Both pairs (5, 6) and (6, 5) are valid solutions since a rectangle can have its length and width interchanged.

The area of a rectangular garage is $30 \, \text{square meters}$ . The perimeter is $22 \, \text{meters}$ . What are the dimensions of the garage? $\newline$ $\_\_\_\_$ meters by $\_\_\_\_$ meters

Full solution

More problems from Area and perimeter: word problems

Related topics

The area of a rectangular garage is 30 square meters30 \, \text{square meters}30square meters. The perimeter is 22 meters22 \, \text{meters}22meters. What are the dimensions of the garage?\newline____\_\_\_\_____ meters by ____\_\_\_\_____ meters

The area of a rectangular garage is $30 \, \text{square meters}$ . The perimeter is $22 \, \text{meters}$ . What are the dimensions of the garage? $\newline$ $\_\_\_\_$ meters by $\_\_\_\_$ meters