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:
[tex]\[ \frac{\text{Length}}{\text{Width}} = \text{Golden Ratio} \][/tex]
Let's denote the length as [tex]\( l \)[/tex] and use the given values to find [tex]\( l \)[/tex]:
[tex]\[ \frac{l}{4 \text{ cm}} = 1.618 \][/tex]
To find [tex]\( l \)[/tex], multiply both sides of the equation by the width:
[tex]\[ l = 4 \text{ cm} \times 1.618 \][/tex]
[tex]\[ l \approx 6.472 \text{ cm} \][/tex]
Now we have the length, but the question asks us to round the answer to the nearest tenth of a centimeter. Rounding [tex]\( 6.472 \text{ cm} \)[/tex] to the nearest tenth gives us [tex]\( 6.5 \text{ cm} \)[/tex].
So, the length of the rectangle to the nearest tenth of a centimeter is [tex]\( 6.5 \text{ cm} \)[/tex].
