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Question 1198079: A rectangle has an area of 300 square centimetres. suppose the width increases from 25 to 30 cm, but the area stays the same.
a) determine the fractional change in the width.
b) determine the fractional change in the length (r) using the constant product principle.
c) Determine the fractional change in the length (r) if the width increases by the fraction, w.
d) Sketch the graph that represents the fractional change in the length as a function of the fractional change in the width.
Answer by onyulee(41)   (Show Source):
You can put this solution on YOUR website! **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!
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