Solve each problem. See Example 1. Michael must build a rectangular storage shed. He wants the length to be 6 ft greater than the width, and the perimeter will be 44 ft. Find the length and the width of the shed.

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Define variables for the dimensions of the shed: let the width be \(w\) feet, and since the length is 6 feet greater than the width, the length will be \(w + 6\) feet.

Write the formula for the perimeter of a rectangle, which is \(P = 2 \times (\text{length} + \text{width})\). Substitute the given perimeter and expressions for length and width: \(44 = 2 \times ((w + 6) + w)\).

Simplify the equation inside the parentheses: \(44 = 2 \times (2w + 6)\).

Distribute the 2 on the right side: \(44 = 4w + 12\).

Solve the linear equation for \(w\) by isolating the variable: subtract 12 from both sides to get \(44 - 12 = 4w\), then divide both sides by 4 to find \(w\). Once \(w\) is found, calculate the length by adding 6 to \(w\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around it, calculated by adding twice the length and twice the width (P = 2L + 2W). Understanding this formula is essential to set up an equation based on the given perimeter.

Algebraic Expressions and Variables

Representing unknown quantities with variables allows us to translate word problems into equations. Here, defining the width as a variable and expressing the length in terms of the width helps form an equation to solve.

Solving Linear Equations

Once the equation is set up, solving linear equations involves isolating the variable to find its value. This process includes combining like terms and performing inverse operations to determine the dimensions of the shed.