To find the length of a golden rectangle when given the width, you can use the golden ratio. The golden ratio (approximately 1.618) is a mathematical ratio that often appears in nature and is used in art, architecture, and design for its aesthetically pleasing properties. For a rectangle to be considered a "golden rectangle," the ratio of its length to its width should be equal to the golden ratio.
The problem gives us the width of the rectangle, which is 4 cm. We want to find the length, so we'll use the formula that comes from the definition of a golden rectangle:
\(\[ \frac{\text{Length}}{\text{Width}} = \text{Golden Ratio} \]\)
Let's denote the length as \(\( l \)\) and use the given values to find \(\( l \)\):
\(\[ \frac{l}{4 \text{ cm}} = 1.618 \]\)
To find \(\( l \)\), multiply both sides of the equation by the width:
\(\[ l = 4 \text{ cm} \times 1.618 \]\)
\(\[ l \approx 6.472 \text{ cm} \]\)
Now we have the length, but the question asks us to round the answer to the nearest tenth of a centimeter. Rounding \(\( 6.472 \text{ cm} \)\) to the nearest tenth gives us \(\( 6.5 \text{ cm} \)\).
So, the length of the rectangle to the nearest tenth of a centimeter is \(\( 6.5 \text{ cm} \)\).
