L3 Area and Volume - Part 1
Completion requirements
Unit 5
Applications of Derivatives
B. Related Rates Problems
Lesson 3: Area and Volume
Have you ever had the feeling that time was running out? When you play a board game with a time limit, does time seem to go faster as time is running out? Does the sand in an hourglass appear to drain faster as the top half gets emptier?
While there is a change in the volume of the sand in each half of the hourglass, there also seems to be a change in the rate at which the sand flows. This is only an illusion. The hole that releases the sand in the hourglass does not change size or shape. The rate at which the sand drains is constant. The changes you do see are a change in volume and a change in the depth of the sand in both halves of the hourglass.
The rate at which the volume of sand is changing depends on the the rate at which the level of the sand is changing. Problems involving these types of rates are called related-rates problems. This kind of problem presents a situation in which one or more quantities are changing with respect to time. The solution to the problem usually requires you to find the rate at which one of the quantities is changing. You will be using the chain rule to help solve these types of problems.
In the next three Lessons, you will study some classic examples of related-rates problems. Using techniques you have studied in previous Units and Lessons, you will solve problems in which the rates of change of some variables are dependent on one another.
While there is a change in the volume of the sand in each half of the hourglass, there also seems to be a change in the rate at which the sand flows. This is only an illusion. The hole that releases the sand in the hourglass does not change size or shape. The rate at which the sand drains is constant. The changes you do see are a change in volume and a change in the depth of the sand in both halves of the hourglass.
The rate at which the volume of sand is changing depends on the the rate at which the level of the sand is changing. Problems involving these types of rates are called related-rates problems. This kind of problem presents a situation in which one or more quantities are changing with respect to time. The solution to the problem usually requires you to find the rate at which one of the quantities is changing. You will be using the chain rule to help solve these types of problems.
In the next three Lessons, you will study some classic examples of related-rates problems. Using techniques you have studied in previous Units and Lessons, you will solve problems in which the rates of change of some variables are dependent on one another.
