Mat1214 Takehome Chapter 3 Section 10 Related Rates

Mat1214 Takehome Chapter 3 Section 10 Related Rates

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Analyze the concepts and methods involved in applying related rates to various problems in physics and engineering, demonstrating understanding through detailed explanations and example calculations.

Introduction

Related rates problems are a fundamental aspect of differential calculus, involving the computation of the rate at which one quantity changes with respect to time, given the rate of change of another related quantity. These problems are common in physics, engineering, and various sciences, providing insights into the dynamic behavior of systems. The core idea is to relate the rates of change of different variables through an equation that models the physical situation, then differentiate implicitly with respect to time to find the unknown rate.

Analysis of Problems

1. Decreasing distance from a boat to a dock

The first problem involves a boat being pulled toward a dock via a rope of which the length is changing at a constant rate. The key variables are the distance from the boat to the dock, the length of the rope, and the rate at which the rope is pulled in. Specifically, the rate of change of the rope length (−2 ft/sec) influences how quickly the boat approaches the dock.

Applying the Pythagorean theorem, if x is the distance from the boat to the dock and l is the length of the rope, then l² = x² + y², where y is a fixed perpendicular distance if applicable. In simpler problem versions, the relationship can be directly between x and l.

By differentiating with respect to time t, using implicit differentiation, one can find the rate at which the distance x decreases when x is 5 ft. The calculations involve plugging in known values and solving for dx/dt, which indicates how fast the boat approaches the dock at that instant.

2. Rate of change of volume in a cylindrical chamber

This problem considers a piston moving into a cylindrical chamber. The radius of the cylinder is fixed at 5 cm, and the piston moves at 8 cm/sec. The goal is to find how fast the volume V of the chamber changes when the piston is 6 cm from the bottom.

The volume of a cylinder is V = πr²h, with r constant. Differentiating with respect to time t yields dV/dt = πr² dh/dt. Substituting the radius and the rate of piston movement provides the rate at which volume increases or decreases.

The challenge is to understand how the changing height translates into volume change, considering the piston’s movement, enabling the calculation of dV/dt at a specific height.

3. Water draining from a conical tank

The third problem involves a conical tank with water draining at a steady rate. The problem provides the rate at which the water level drops (3 ft/min) and asks to find the rate at which water volume is leaving the tank when the water depth is 4 ft.

The volume of a cone V = (1/3)πr²h depends on the water radius and height. Because the tank's shape is conical, the radius at any water level relates to the height, typically through similar triangles. Differentiating V with respect to time involves both r and h, but because r varies proportionally with h, the relationship simplifies to express dr/dt in terms of dh/dt.

By plugging in the known values and the relationships, the rate of volume drainage can be computed, illustrating the application of related rates to fluid dynamics in tanks.

Discussion

These three problems exemplify how related rates involve understanding the geometry of the physical system, establishing relationships among variables, and applying implicit differentiation. By differentiating relevant equations with respect to time, one can determine how different quantities evolve simultaneously. These problem types are vital in real-world applications, such as navigation, engineering design, and fluid mechanics.

In the first problem, the key concept is the Pythagorean theorem and its dynamic application. In the second, understanding volume change in a cylinder with fixed radius. The third highlights the principles of similar triangles and volumetric calculations for tanks. All require translating physical relationships into mathematical models and differentiating appropriately.

Conclusion

Applying related rates effectively requires careful identification of variables, establishing relevant geometric or physical relationships, and executing implicit differentiation. Mastery of these techniques allows for solving diverse problems involving rates of change in physics, engineering, and beyond. Such problems demonstrate the interconnectedness of calculus concepts and their practical importance in analyzing dynamic systems.

References

  • Anton, H., Bivens, I., & Davis, S. (2016). Calculus: Early Transcendentals. John Wiley & Sons.
  • Larson, R., & Edwards, B. H. (2016). Calculus. Cengage Learning.
  • Salas, S., Hille, E., & Etgen, G. J. (2013). Calculus: One and Several Variables. Pearson.
  • Thomas, G. B., & Finney, R. L. (2002). Calculus and Analytic Geometry. Addison Wesley.
  • Stewart, J. (2015). Calculus: Concepts and Contexts. Cengage Learning.
  • Swokowski, E., & Cole, J. (2011). Calculus with Applications. Cengage Learning.
  • Taylor, J. R. (2005). Classical Mechanics. University Science Books.
  • Jeffrey, A., & Waller, J. (2019). Fluid Mechanics. Oxford University Press.
  • Rinehart, R., & Rudy, J. (2020). Engineering Calculations in Fluid Dynamics. Springer.
  • Strang, G. (2016). Calculus. Wellesley-Cambridge Press.

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