Related Rates Examples

Related rates show how different quantities change together over time. By applying the chain rule and implicit differentiation, we can analyze real-world scenarios, like expanding circles or moving objects, to understand their interconnected rates of change.

1. Expanding circle (radius and area)

   • The area ( A ) of a circle is given by the formula ( A = \pi r^2 ), where ( r ) is the radius.
   • As the radius changes over time, the rate of change of the area can be found using the chain rule: ( \frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt} ).
   • This example illustrates how a small change in radius can lead to a significant change in area.

2. Filling container (volume and height)

   • The volume ( V ) of a container (like a cylinder) is calculated using ( V = \pi r^2 h ), where ( r ) is the radius and ( h ) is the height.
   • When filling the container, the rate of change of volume relates to the rate of change of height: ( \frac{dV}{dt} = \pi r^2 \cdot \frac{dh}{dt} ).
   • Understanding this relationship helps in determining how quickly the height of the liquid rises as it is poured in.

3. Ladder sliding down a wall

   • The relationship between the height of the ladder on the wall, the distance from the wall, and the length of the ladder can be described using the Pythagorean theorem: ( x^2 + y^2 = L^2 ).
   • As the ladder slides down, the rates of change of ( x ) (distance from the wall) and ( y ) (height on the wall) can be related through implicit differentiation.
   • This scenario emphasizes the connection between geometric constraints and rates of change.

4. Shadow length of a moving object

   • The length of a shadow can be modeled using similar triangles, relating the height of the object and the angle of the light source.
   • As the object moves, the rate of change of the shadow length can be determined by differentiating the relationship between the height and the distance from the light source.
   • This example highlights how angles and distances can affect rates of change in real-world scenarios.

5. Water draining from a cone

   • The volume ( V ) of a cone is given by ( V = \frac{1}{3} \pi r^2 h ), where ( r ) is the radius and ( h ) is the height.
   • As water drains, both the height and radius change, and their rates can be related through the volume formula.
   • This example illustrates how changing dimensions affect the volume and the rate at which it decreases.

6. Distance between two moving objects

   • The distance ( d ) between two objects can be expressed as a function of their respective positions over time.
   • By differentiating the distance function, we can find the rate at which the distance between the two objects is changing.
   • This scenario is useful for understanding relative motion and how it impacts distance.

7. Changing angle of a triangle

   • In a triangle, the relationship between the sides and angles can be described using trigonometric functions.
   • As one angle changes, the rates of change of the sides can be determined using implicit differentiation.
   • This example emphasizes the interconnectedness of angles and side lengths in geometric figures.

8. Growing snowball (volume and radius)

   • The volume ( V ) of a sphere is given by ( V = \frac{4}{3} \pi r^3 ), where ( r ) is the radius.
   • As the snowball grows, the rate of change of volume can be related to the rate of change of radius using the chain rule: ( \frac{dV}{dt} = 4\pi r^2 \cdot \frac{dr}{dt} ).
   • This example illustrates how a small change in radius can lead to a significant increase in volume.

9. Inflating balloon (volume and radius)

   • The volume ( V ) of a balloon can be modeled as a sphere: ( V = \frac{4}{3} \pi r^3 ).
   • As the balloon inflates, the relationship between the rate of change of volume and radius can be expressed using the chain rule.
   • This scenario highlights the rapid increase in volume as the radius expands.

10. Car approaching an intersection

    • The distance between the car and the intersection can be modeled as a function of time as the car approaches.
    • By differentiating this distance function, we can find the rate at which the car is approaching the intersection.
    • This example is crucial for understanding motion and the implications of speed in relation to time and distance.