Let's solve the given problem step by step.

1. Forming the equations:

   The given equation of the rectangular hyperbola is: $y = \frac{4}{x}$

   One curved road passes through the point (2, 2), and the other passes through (-2, -2). Both points satisfy the hyperbola equation: $y = \frac{4}{x}$

   For the new straight road represented by a line passing through the points (0, 4) and (8, 0), we need to find its equation. The slope $m$ of the line passing through (0, 4) and (8, 0) is: $m = \frac{0 - 4}{8 - 0} = -\frac{1}{2}$

   Using the point-slope form $y = mx + c$ with the point (0, 4), we get: $y = -\frac{1}{2}x + 4$

2. Finding points of intersection:

   We need to find the points of intersection of the line $y = -\frac{1}{2}x + 4$ with the hyperbola $y = \frac{4}{x}$.

   Set the equations equal to each other: $-\frac{1}{2}x + 4 = \frac{4}{x}$

   Multiply both sides by $x$ to clear the fraction: $-\frac{1}{2}x^2 + 4x = 4$

   Rearrange to form a quadratic equation: $-\frac{1}{2}x^2 + 4x - 4 = 0$

   Multiply through by -2 to clear the fraction: $x^2 - 8x + 8 = 0$

   Solve the quadratic equation using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$: $x = \frac{8 \pm \sqrt{64 - 32}}{2} = \frac{8 \pm \sqrt{32}}{2} = \frac{8 \pm 4\sqrt{2}}{2} = 4 \pm 2\sqrt{2}$

   So, the $x$-coordinates of the points of intersection are: $x = 4 + 2\sqrt{2} \quad \text{and} \quad x = 4 - 2\sqrt{2}$

   Substitute these back into the line equation to find the $y$-coordinates: $y = -\frac{1}{2}(4 + 2\sqrt{2}) + 4 = 2 - \sqrt{2}$ $y = -\frac{1}{2}(4 - 2\sqrt{2}) + 4 = 2 + \sqrt{2}$

   Therefore, the points of intersection are: $\left(4 + 2\sqrt{2}, 2 - \sqrt{2}\right) \quad \text{and} \quad \left(4 - 2\sqrt{2}, 2 + \sqrt{2}\right)$

3. Interpret and present your solutions:

   These points represent where the new straight road intersects the existing curved roads. Specifically:

   • $\left(4 + 2\sqrt{2}, 2 - \sqrt{2}\right)$
   • $\left(4 - 2\sqrt{2}, 2 + \sqrt{2}\right)$

   This information is important for city planning and construction, ensuring that the new straight road properly connects with the existing curved roads at the specified points.

Would you like further details or have any questions?

Here are 8 relative questions that expand on this information:

1. How would the intersection points change if the slope of the new road were different?
2. What are the properties of a rectangular hyperbola?
3. How can you derive the equation of a line given two points?
4. What role do intersection points play in real-world applications like road construction?
5. How can we use calculus to analyze the curvature of the roads at the intersection points?
6. Can you find the distance between the two intersection points?
7. How does changing the location of points through which the line passes affect the intersection points?
8. What are the potential real-world challenges in constructing a road that intersects curved roads?

Tip: Always verify solutions by plugging them back into the original equations to ensure accuracy.