The length of one side of a golden rectangle is 7.7 meters. Find the length of the other side. Notice that the other side could be either longer or shorter than the given side. Use the approximation $\phi \approx 1.62$ for your work.
If 7.7 is the long side, the short side is $\square$ meters.
(Round to two decimal places as needed.)
If 7.7 is the short side, the long side is $\square$ meters.
(Round to two decimal places as needed.)
Answer
Final Answer: If 7.7 meters is the long side, the short side is approximately \(\boxed{4.75}\) meters. If 7.7 meters is the short side, the long side is approximately \(\boxed{12.47}\) meters.
Step 1 ：The golden ratio, often denoted by the Greek letter φ (phi), is an irrational mathematical constant approximately equal to 1.6180339887. In a golden rectangle, the ratio of the longer side (length) to the shorter side (width) is the golden ratio.
Step 2 ：Given that one side of the rectangle is 7.7 meters, we can find the length of the other side by either dividing or multiplying by the golden ratio, depending on whether the given side is the longer or shorter side.
Step 3 ：If 7.7 meters is the long side, the short side can be found by dividing 7.7 by the golden ratio. Using the approximation \(\phi \approx 1.62\), we get \(\frac{7.7}{1.62} \approx 4.75\) meters.
Step 4 ：If 7.7 meters is the short side, the long side can be found by multiplying 7.7 by the golden ratio. Using the approximation \(\phi \approx 1.62\), we get \(7.7 \times 1.62 \approx 12.47\) meters.
Step 5 ：Final Answer: If 7.7 meters is the long side, the short side is approximately \(\boxed{4.75}\) meters. If 7.7 meters is the short side, the long side is approximately \(\boxed{12.47}\) meters.
