1. Section 4.6 Related Rates

2. We have seen that if we can model a situation by a continuous function, we can use calculus to analyze changes that have taken place
We have been concerned with how the rate of change in the output variable is affected by a change in the input variable
Now we will look at the interaction of the rates of change of input and output variable with respect to a third variable
The interconnection of the input and output variables is reflected in an interaction between their rates with respect to a third variable

3. Consider the following problem:
A spherical balloon of radius r centimeters has a volume given by
Find dV/dr when r = 1 , r = 2 and r = 3 and give a practical interpretation of your answers.
Suppose that the balloon is being inflated in such a way that r(t) = 2t centimeters after t seconds. How fast is the volume of the balloon increasing when r = 1? When r = 2?
Air is being blown in the balloon at a constant rate of 50 cubic centimeters per second. How fast is the radius of the balloon increasing when r = 1? When r = 2?
In order to answer the last two questions we need a technique called implicit differentiation

4. Implicit Differentiation
Implicit differentiation involves differentiating both sides with respect to a third variable and then solving for the derivative
Take our function
We will take the derivative of both sides with respect to t (time) using the chain rule
Then we can look at finding how the radius is changing

5. A spherical balloon of radius r centimeters has a volume given by
Suppose that the balloon is being inflated in such a way that r(t) = 2t centimeters after t seconds. How fast is the volume of the balloon increasing when r = 1? When r = 2?
Air is being blown in the balloon at a constant rate of 50 cubic centimeters per second. How fast is the radius of the balloon increasing when r = 1? When r = 2?

6. 5 Steps to solving related rates problems
Read problem and identify variables involved. Identify independent and all dependent variables.
Use the given equation (or find one) relating the variables.
Differentiate both sides of the equation with respect to the independent variable.
Substitute in the known quantities and rates into the related rates equation and solve for the unknown rate
Interpret in the context of the problem

7. A ladder 10 feet long rests against a vertical wall
A ladder 10 feet long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?

8. A water tank has the shape of cylinder with base radius 2m and height 4m. If water is being pumped into the tank at a rate of 2m3/min, find the rate at which the water level is rising when the water is 3m deep.

9. A water tank has the shape of an inverted circular cone with base radius 2m and height 4m. If water is being pumped into the tank at a rate of 2m3/min, find the rate at which the water level is rising when the water is 3m deep.