Not who, but what? What are related rates?
Related Rates - The process of finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known
Steps for Solving a Related Rates Problem
1. Draw a visual for the situation, but include only what is not changing
2. Identify all the information you know and what you are finding
3. Identify an equation that you can take the derivative of
4. Make sure everything that is changing is represented by a variable
5. Solve for the variable you are looking for
Taking the derivative in respect time...
x^2 = 2x dx/dt 2x^4 = 8x^3 dx/dt
Quick Tips:
-On the visual, only include what is not changing, like the length of a ladder or radius of a sphere
- Assign everything that changes a variable so it is easier to keep track of when taking the derivative
Video Lessons
Example 1
Water is draining out of a cylindrical tank at a rate of 3 ft^3/min. The diameter of the tank is 6 ft and the tank is 8 ft tall. How fast is the water level falling when the water is 4 ft deep?
https://www.youtube.com/watch?v=uxHCE2HoLA
Example 2
A 12 foot ladder attatched to a firetruck is leaning against a burning building. The top of the ladder is sliding down the wall at a rate of 3ft/sec. How fast is the bottom of the ladder moviing along the ground when the bottom of the ladder is 4 ft from the wall?
www.educreations.com/lesson/view/related-rates-one/42284459/?s=H1ZO4l&ref=appemail
Example 3
Gravel is being dumped from a conveyor belt at a rate of 20 ft^3/min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?
www.educreations.com/lesson/view/related-rates-3/42299177/?s=nu44CL&ref=appemail
Example 4
A conical fishtank is 100 ft across the top and 150ft deep. If water flows into the tank at a rate of 20 cubic feet per hour, find the rate of change of depth when the water is 60 ft deep.
www.educreations.com/lesson/view/related-rates-two/42284474/?s=SjkFKg&ref=appemail
Example 5
When a circular of silver is heated over a fire its radius is increasing at a rate of 1/4 cm/sec. At what rate is the sheilds area increasing when the radius is 20 cm?
www.youtube.com/watch?v=2a-n6KovwxE
Example 6
A spherical balloon’s radius is increasing at a rate of ⅕ cm/(s^2). At what rate is the balloon expanding when the radius is 7 cm? Given: A=4Pi(r^2)
www.educreations.com/lesson/view/related-rates-6/42299304/?s=EJiQNW&ref=appemail
Example 7 and 8
www.educreations.com/lesson/view/related-rates/42253168/?s=QvY9Z9&ref=app
Practice Problems
1.
A conical cup is 4 inches across the top, 6 inches deep and is full of gatorade. Suddenly, the cup begins to leak and spills the gatorade at a rate of 2 cubic inches per meter. How fast is the liquid level dropping when the gatorade is 1.5 inches deep?
1. Draw a visual for the situation, but include only what is not changing
2. Identify all the information you know and what you are finding
3. Identify an equation that you can take the derivative of
4. Make sure everything that is changing is represented by a variable
5. Solve for the variable you are looking for
Taking the derivative in respect time...
x^2 = 2x dx/dt 2x^4 = 8x^3 dx/dt
Quick Tips:
-On the visual, only include what is not changing, like the length of a ladder or radius of a sphere
- Assign everything that changes a variable so it is easier to keep track of when taking the derivative
Video Lessons
Example 1
Water is draining out of a cylindrical tank at a rate of 3 ft^3/min. The diameter of the tank is 6 ft and the tank is 8 ft tall. How fast is the water level falling when the water is 4 ft deep?
https://www.youtube.com/watch?v=uxHCE2HoLA
Example 2
A 12 foot ladder attatched to a firetruck is leaning against a burning building. The top of the ladder is sliding down the wall at a rate of 3ft/sec. How fast is the bottom of the ladder moviing along the ground when the bottom of the ladder is 4 ft from the wall?
www.educreations.com/lesson/view/related-rates-one/42284459/?s=H1ZO4l&ref=appemail
Example 3
Gravel is being dumped from a conveyor belt at a rate of 20 ft^3/min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?
www.educreations.com/lesson/view/related-rates-3/42299177/?s=nu44CL&ref=appemail
Example 4
A conical fishtank is 100 ft across the top and 150ft deep. If water flows into the tank at a rate of 20 cubic feet per hour, find the rate of change of depth when the water is 60 ft deep.
www.educreations.com/lesson/view/related-rates-two/42284474/?s=SjkFKg&ref=appemail
Example 5
When a circular of silver is heated over a fire its radius is increasing at a rate of 1/4 cm/sec. At what rate is the sheilds area increasing when the radius is 20 cm?
www.youtube.com/watch?v=2a-n6KovwxE
Example 6
A spherical balloon’s radius is increasing at a rate of ⅕ cm/(s^2). At what rate is the balloon expanding when the radius is 7 cm? Given: A=4Pi(r^2)
www.educreations.com/lesson/view/related-rates-6/42299304/?s=EJiQNW&ref=appemail
Example 7 and 8
www.educreations.com/lesson/view/related-rates/42253168/?s=QvY9Z9&ref=app
Practice Problems
1.
A conical cup is 4 inches across the top, 6 inches deep and is full of gatorade. Suddenly, the cup begins to leak and spills the gatorade at a rate of 2 cubic inches per meter. How fast is the liquid level dropping when the gatorade is 1.5 inches deep?
2.
A balloon is floating in the sky, 200 ft above the ground. Susie is holding the balloon, but it is being blown away at 20 ft/sec. At what rate is the string being let out when the string is already 400 ft in length?
A balloon is floating in the sky, 200 ft above the ground. Susie is holding the balloon, but it is being blown away at 20 ft/sec. At what rate is the string being let out when the string is already 400 ft in length?
3.
A ladder is 25 ft long and leaning against the wall of a shed. The base of the ladder is pulled away from the wall at a rate of 1 ft/sec. How fast is the top of the ladder moving when the base is 10 ft from the wall?
A ladder is 25 ft long and leaning against the wall of a shed. The base of the ladder is pulled away from the wall at a rate of 1 ft/sec. How fast is the top of the ladder moving when the base is 10 ft from the wall?
4.
Air is being pumped into a soccer ball at the rate of 5ft^3 per minute. Find the rate of change of the radius when the radius is 3ft.
Air is being pumped into a soccer ball at the rate of 5ft^3 per minute. Find the rate of change of the radius when the radius is 3ft.
5.
A 10 ft. ladder is leaning against a house when the base starts to slide away. When the base is 8 ft. from the house, it was moving at a rate of 3 ft/sec. How fast is the top of the ladder sliding down the wall at that moment?
A 10 ft. ladder is leaning against a house when the base starts to slide away. When the base is 8 ft. from the house, it was moving at a rate of 3 ft/sec. How fast is the top of the ladder sliding down the wall at that moment?
6.
Water is pumped into a conical reservior at a rate of 6m^3/min. The reservior is 12 ft across and 14 ft deep. Find the rate of change when the water is 8 ft deep.
Water is pumped into a conical reservior at a rate of 6m^3/min. The reservior is 12 ft across and 14 ft deep. Find the rate of change when the water is 8 ft deep.
7.
A boat is being pulled into a dock by a rope at a rate of .5 ft/sec. The dock is 8 ft above the boat. At what rate is the angle between the rope and the water changing when there is 20 ft of rope out?
A boat is being pulled into a dock by a rope at a rate of .5 ft/sec. The dock is 8 ft above the boat. At what rate is the angle between the rope and the water changing when there is 20 ft of rope out?
8.
A jet ski is 50 miles north of a boat and is gliding due south at 25 mph. The boat is sailing due east at 10 mph. At what rate is the distance between them changing at the end of hour two.
A jet ski is 50 miles north of a boat and is gliding due south at 25 mph. The boat is sailing due east at 10 mph. At what rate is the distance between them changing at the end of hour two.
9.
A 10 foot tall ladder is resting against a wall the bottom is initially 3 foot from the wall and is being pushed out from the wall at 1/3 ft./sec.2 how fast is the top of the ladder sliding down the wall after 5 seconds.
A 10 foot tall ladder is resting against a wall the bottom is initially 3 foot from the wall and is being pushed out from the wall at 1/3 ft./sec.2 how fast is the top of the ladder sliding down the wall after 5 seconds.
Resources
http://tutorial.math.lamar.edu/Classes/CalcI/RelatedRates.aspx
http://17calculus.com
http://www.mathscoop.com/calculus/derivatives/applications/related-rates-cone-problems.php
http://tutorial.math.lamar.edu/Classes/CalcI/RelatedRates.aspx
http://17calculus.com
http://www.mathscoop.com/calculus/derivatives/applications/related-rates-cone-problems.php