Find the exact solution to each problem. If the exact...

Find the exact solution to each problem. If the exact solution is an irrational number, then also find an approximate decimal solution. See Examples 7 and 8. Golden Rectangle. One principle used by the ancient Greeks to get shapes that are pleasing to the eye in art and architecture was the Golden Rectangle. If a square is removed from one end of a Golden Rectangle, as shown in the figure, the sides of the remaining rectangle are proportional to the original rectangle. So the length and width of the original rectangle satisfy $$\frac{L}{W}=\frac{W}{L-W}$$ CAN'T COPY THE FIGURE If the length of a Golden Rectangle is 10 meters, then what is its width?

Find the exact solution to each problem. If the exact solution is an irrational number, then also find an approximate decimal solution. See Examples 7 and 8.
Golden Rectangle. One principle used by the ancient Greeks to get shapes that are pleasing to the eye in art and architecture was the Golden Rectangle. If a square is removed from one end of a Golden Rectangle, as shown in the figure, the sides of the remaining rectangle are proportional to the original rectangle. So the length and width of the original rectangle satisfy
$$\frac{L}{W}=\frac{W}{L-W}$$
CAN'T COPY THE FIGURE
If the length of a Golden Rectangle is 10 meters, then what is its width?

Key Concepts

Algebraic Equation Solving

Solving equations derived from proportional relationships typically involves techniques such as cross-multiplication and rearrangement. In geometric contexts, these methods often lead to quadratic equations, where one must find the exact values of unknown dimensions before considering any approximations.

Golden Ratio

The Golden Ratio is a special number often denoted by ? (phi), which appears when a line or rectangle is divided in such a way that the ratio of the whole to the larger part is the same as the ratio of the larger part to the smaller. This concept is widely used in art, architecture, and nature due to its aesthetically pleasing properties and its unique mathematical definition.

Proportional Relationships

Proportional relationships involve setting two ratios equal to each other, which is a common way to express similarity or scaling in geometry. In problems like the golden rectangle, the consistent ratio between dimensions leads to an equation that relates different parts of the shape, providing a pathway to solve for unknown quantities.

Irrational Numbers and Approximation Techniques

When the exact solution of an equation results in an irrational number—a non-repeating, non-terminating decimal—it is important to both retain the exact algebraic form and provide a numerical approximation for practical interpretation. This dual approach ensures precision in theory while making the result accessible for applications.

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