Let the total length of the rope be $L$, the length of the smaller part be $x$, and the length of the larger part be $L - x$.

We are given that the ratio of the length of the whole rope to the smaller part is equal to the square of the ratio of the lengths of the larger and smaller parts. This can be mathematically expressed as:

$\frac{L}{x} = \left( \frac{L - x}{x} \right)^2$

Step 1: Simplify the equation

First, rewrite the equation as:

$\frac{L}{x} = \frac{(L - x)^2}{x^2}$

Multiplying both sides by $x^2$ to eliminate the denominators:

$Lx = (L - x)^2$

Step 2: Expand and simplify

Expand the right-hand side:

$Lx = L^2 - 2Lx + x^2$

Rearrange the equation to move all terms to one side:

$0 = L^2 - 3Lx + x^2$

Step 3: Solve the quadratic equation

This is a quadratic equation in $x$. Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, we identify $a = 1$, $b = -3L$, and $c = L^2$. Substituting these values into the quadratic formula:

$x = \frac{-(-3L) \pm \sqrt{(-3L)^2 - 4(1)(L^2)}}{2(1)}$

Simplifying further:

$x = \frac{3L \pm \sqrt{9L^2 - 4L^2}}{2}$ $x = \frac{3L \pm \sqrt{5L^2}}{2}$ $x = \frac{3L \pm L\sqrt{5}}{2}$

Since $x$ is the smaller part, we take the negative root:

$x = \frac{3L - L\sqrt{5}}{2}$ $x = L \cdot \frac{3 - \sqrt{5}}{2}$

Step 4: Approximate the value

Given that $\sqrt{5} = 2.236$, we substitute:

$x = L \cdot \frac{3 - 2.236}{2}$ $x = L \cdot \frac{0.764}{2}$ $x \approx L \cdot 0.382$

Step 5: Find the fraction

The fraction of the smaller part $x$ to the bigger part $L - x$ is approximately:

$\frac{x}{L - x} = \frac{0.382L}{1 - 0.382}L = \frac{0.382}{0.618} \approx 0.618$

Therefore, the fraction of the length of the bigger part is approximately $\frac{3}{5}$.

Would you like further details or have any questions?

Here are 5 related questions to think about:

1. What is the total length of the rope if the smaller part is 10 meters long?
2. How does the formula change if the ratio is a cube instead of a square?
3. Can you derive the same result geometrically instead of algebraically?
4. How would the solution change if the rope was cut into three parts with a similar condition?
5. What happens to the ratio if the rope’s length changes but the square ratio condition remains?

Tip: Always check if symmetry can simplify your problem, especially in ratio-related problems.