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The length of a rectangle is 4 inches longer than it is wide. If the area is 32 square inches, what are the dimensions of the rectangle? The width, or shorter side is The length, or longer side is \( \square \) inches

Ask by Bolton Ross. in the United States
Dec 06,2024

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Answer

Width: 4 inches, Length: 8 inches

Solution

Let's denote the width of the rectangle as \( w \) inches. Given that the length of the rectangle is 4 inches longer than its width, we can express the length as \( w + 4 \) inches. The area of a rectangle is given by the formula: \[ \text{Area} = \text{Length} \times \text{Width} \] Given that the area of the rectangle is 32 square inches, we can set up the equation: \[ 32 = (w + 4) \times w \] Now, we can solve this equation to find the dimensions of the rectangle. Solve the quadratic equation by following steps: - step0: Solve by factoring: \(32=\left(w+4\right)w\) - step1: Multiply the terms: \(32=w\left(w+4\right)\) - step2: Swap the sides: \(w\left(w+4\right)=32\) - step3: Expand the expression: \(w^{2}+4w=32\) - step4: Move the expression to the left side: \(w^{2}+4w-32=0\) - step5: Factor the expression: \(\left(w-4\right)\left(w+8\right)=0\) - step6: Separate into possible cases: \(\begin{align}&w-4=0\\&w+8=0\end{align}\) - step7: Solve the equation: \(\begin{align}&w=4\\&w=-8\end{align}\) - step8: Rewrite: \(w_{1}=-8,w_{2}=4\) The solutions to the equation are \( w = -8 \) and \( w = 4 \). Since the width cannot be negative, the width of the rectangle is 4 inches. Now, we can find the length of the rectangle by adding 4 inches to the width: \[ \text{Length} = 4 + 4 = 8 \text{ inches} \] Therefore, the dimensions of the rectangle are: - Width: 4 inches - Length: 8 inches

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To solve for the dimensions of the rectangle, we start by letting the width be \( w \) inches. Then, the length would be \( w + 4 \) inches. The area of the rectangle can be expressed as: \[ w(w + 4) = 32 \] Expanding this gives: \[ w^2 + 4w - 32 = 0 \] Using the quadratic formula, we solve for \( w \). The width comes out to be 4 inches, and substituting this back, the length is 8 inches. Now you know that the width is 4 inches and the length is 8 inches! Did you know that the concept of area dates back to ancient civilizations? The Egyptians used simple formulas for rectangles hundreds of years ago to calculate land area for agriculture. This historical perspective shows how foundational geometric concepts are to engineering and architecture, still applied today! In practical scenarios, converting measurements can sometimes cause confusion. Always double-check that you're working with the same units, especially when dealing with properties like area or volume. A common mistake is mixing inches and feet, which can lead to calculation errors. Standardizing units helps in avoiding these pitfalls!

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