Any equation involving two or more variables that are differentiable functions of time "t" can be used to find an equation that relates their corresponding rates. Finding related rate equations are quite simple, set the derivative in respect to "t" and them differentiate implicitly.
For example the equation for finding the volume of a sphere becomes dv/dt and instead of (4/3)(pi)r^3 it becomes equal to 4(pi)r^2(dr/dt).
When solving for related rate problems
1. Understand the problem
2. Develop a mathematical model of the problem
3. Write an equation relating the variable whose rate of change you seek with the variable(s) whose rate of change you know
4. Differentiate both sides of the equation implicitly with respect to time "t"
5. Substitute values for any quantities that depend on times
6. Interpret the solution
You will be given enough information to find the answer but finding out what you need to find out might be a challenge, it may help to create a table of "things you know" and "things you want to know" or "changing" and "not changing" variables in order to plug them back into the equation in respect to "t" to find what you want to know.
An example of related rate problem would be a police car approaching an intersection chasing a speeding car that has already turned a corner and is headed East. When the police car is 0.6mi North of the intersection and the speeding car is 0.8mi East , the distance between them is changing at 20mph. If the police car is going 60mph how fast is the other car going?
First, draw a picture of the intersection and figure out your compass with the location of where your police car and speeding car is. We find that finding the distance between the two cars can be found by using the Pythagorean theorem. With the diagram, we want to find dx/dt when y=0.6mi and x=0.8mi with the starting equation of y^2+x^2=z^2.
For example the equation for finding the volume of a sphere becomes dv/dt and instead of (4/3)(pi)r^3 it becomes equal to 4(pi)r^2(dr/dt).
When solving for related rate problems
1. Understand the problem
2. Develop a mathematical model of the problem
3. Write an equation relating the variable whose rate of change you seek with the variable(s) whose rate of change you know
4. Differentiate both sides of the equation implicitly with respect to time "t"
5. Substitute values for any quantities that depend on times
6. Interpret the solution
You will be given enough information to find the answer but finding out what you need to find out might be a challenge, it may help to create a table of "things you know" and "things you want to know" or "changing" and "not changing" variables in order to plug them back into the equation in respect to "t" to find what you want to know.
An example of related rate problem would be a police car approaching an intersection chasing a speeding car that has already turned a corner and is headed East. When the police car is 0.6mi North of the intersection and the speeding car is 0.8mi East , the distance between them is changing at 20mph. If the police car is going 60mph how fast is the other car going?
First, draw a picture of the intersection and figure out your compass with the location of where your police car and speeding car is. We find that finding the distance between the two cars can be found by using the Pythagorean theorem. With the diagram, we want to find dx/dt when y=0.6mi and x=0.8mi with the starting equation of y^2+x^2=z^2.
Here is another example.
Need extra help? Check out this video on cones
Or this video about hot air balloons
Or this video on falling ladders
Related rates are so applicable in real life, huh?
Or this video about hot air balloons
Or this video on falling ladders
Related rates are so applicable in real life, huh?