Question 1198079
**a) Determine the fractional change in the width.**

* **Original Width:** 25 cm
* **New Width:** 30 cm
* **Change in Width:** 30 cm - 25 cm = 5 cm
* **Fractional Change in Width:** (Change in Width) / (Original Width) = 5 cm / 25 cm = 1/5 or 0.20

**b) Determine the fractional change in the length (r) using the constant product principle.**

* **Constant Product Principle:** For a constant area, if one dimension increases, the other dimension must decrease proportionally.

* **Original Area:** 300 cm²
* **Original Width:** 25 cm
* **Original Length:** 300 cm² / 25 cm = 12 cm

* **New Width:** 30 cm
* **New Length:** 300 cm² / 30 cm = 10 cm

* **Change in Length:** 10 cm - 12 cm = -2 cm
* **Fractional Change in Length (r):** (-2 cm) / 12 cm = -1/6 or -0.1667

**c) Determine the fractional change in the length (r) if the width increases by the fraction, w.**

* Let 'w' be the fractional increase in width.
* New Width = Original Width * (1 + w)

* Since Area = Width * Length:
    * Original Width * Original Length = New Width * New Length
    * Original Width * Original Length = (Original Width * (1 + w)) * New Length
    * New Length = (Original Length) / (1 + w)

* Fractional Change in Length (r) = (New Length - Original Length) / Original Length
    * r = [(Original Length) / (1 + w) - Original Length] / Original Length
    * r = [1 / (1 + w) - 1] 
    * r = [1 - (1 + w)] / (1 + w)
    * r = -w / (1 + w)

**d) Sketch the graph that represents the fractional change in the length as a function of the fractional change in the width.**

* This equation (r = -w / (1 + w)) represents a hyperbola. 
* Key features of the graph:
    * As 'w' (fractional change in width) increases, 'r' (fractional change in length) becomes increasingly negative.
    * When 'w' is 0 (no change in width), 'r' is also 0.
    * The graph will approach an asymptote at 'w' = -1, as the length would become infinitely large.

**In Summary:**

* The fractional change in the width is 0.20.
* The fractional change in the length is -0.1667.
* The general relationship between the fractional change in width (w) and the fractional change in length (r) is given by: r = -w / (1 + w).
* The graph of this relationship is a hyperbola.

I hope this comprehensive explanation is helpful!