2a rectangle's width is three times its height, and its perimiter is 120 mm . Find the rectangle's width and its height.
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To solve this puzzle, let’s first define the rectangle's height as \( h \). Since the width is three times the height, we can express the width as \( 3h \). The formula for the perimeter of a rectangle is \( P = 2(\text{width} + \text{height}) \). Plugging in our expressions gives us: \[ 120 = 2(3h + h) \] This simplifies to: \[ 120 = 2(4h) \] \[ 120 = 8h \] \[ h = 15 \] Now that we have the height, we can find the width: \[ \text{width} = 3h = 3 \times 15 = 45 \] So, the rectangle's height is 15 mm, and its width is 45 mm! A fun fact: rectangles with such proportional dimensions are often used in design for their pleasing aesthetic. Many logos and artworks use the golden ratio, which takes inspiration from similar ratios! Speaking of ratios, it’s important in math to keep track of units and proportions. Always double-check your ratios and calculations, as common mistakes can happen when setting up equations. Mathematically suspecting the relationships between dimensions can guide you to the answer faster and prevent confusion!
