Related Rates - Conical Tank, Ladder Angle & Shadow Problem, Circle & Sphere - Calculus

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This calculus video tutorial explains how to solve related rates problems using derivatives. It shows you how to calculate the rate of change with respect to radius, height, surface area, or volume of a sphere, circle, cone, etc. This video contains plenty of examples and practice problems such as the inverted conical tank problem, the ladder angle problem, similar triangle shadow problem, problems with circles, spheres, cubes, cones, squares, and triangles and so forth.

Related Rates - Free Formula Sheet:


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Introduction to Limits:


Derivatives - Fast Review:


Introduction to Related Rates:


Derivative Notations:


Related Rates - The Cube:


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Inflated Balloon & Melting Snowball:


Gravel Dumped Into Conical Tank:


Related Rates - Area of a Triangle:


Related Rates - The Ladder Problem:


Related Rates - The Distance Problem:


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Related Rates - Airplane Problems:


Related Rates - The Shadow Problem:


Related Rates - The Baseball Diamond Problem:


Related Rates - The Angle of Elevation Problem:


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Final Exams and Video Playlists:


Full-Length Videos and Worksheets:



Here is a list of problems.
1. Finding dx/dt, dy/dt and dz/dt - Pythagorean Theorem - Right Triangle Trigonometry
2. How to find the rate of change of the distance between the origin and a moving point on the graph if the y-coordinate is increasing
3. The radius of a circle is decreasing at a rate of 4cm/min. How fast is the area and circumference changing when the radius is 8cm?
4. The surface area of a snowball at a rate of 6 square feet per hour, how fast is the diameter changing when the radius is 2 ft?
5. The side length of a square increases at a rate of 3 inches per second, how fast is the area and perimeter of the square changing when the side length is 5 in ?
6. A spherical balloon is inflated with gas at a rate of 900 cubic centimeters per minute (cm^3/min), how fast is the radius of the balloon changing?
7. The side lengths of a cube are increasing at a rate of 5 cm/s, How fast is the surface area and volume increasing?
8. A 13 foot ladder leans against a house. The ladder slides down the wall at a rate of 3 ft/min. How fast is the ladder moving away from the base of the wall when the foot of the ladder is currently 5ft from the wall? How fast is the area of the triangle changing? How fast is the angle between the ground and ladder changing?
9. Gravel is being dumped from a conveyor belt at a rate of 100 cubic feet per min (ft^3/min) forming a conical pile whose base diameter is two times the altitude. How fast is the height changing?
10. Water is leaking out of an inverted conical tank at 500 cm^3/min. The tank has a height of 24 cm and a radius of 6cm. Find the rate at which water is being poured into the tank if the water level is rising at 15cm/min.
11. A street light is mounted on a pole 24 ft tall. A man 6ft tall walks away from the pole at a rate of 4ft/s. How fast is the tip of his shadow moving when he is 20ft from the pole? How fast is the length of his shadow changing at this instant?
12. A spotlight shines on a wall 18m away. If a 2m tall man walks toward the building at a speed of 2m/s, how fast is the length of his shadow on the building changing?
13. Two cars are moving starting from the same point.
14. At 1:00pm, ship B is 150 miles from ship A. Ship A is moving 30mph north and ship B is moving 20mph south. How fast is the distance changing at 3:00pm?
15. Airplane Problem - Travels Horizontally at an altitude of 3 miles. Radar Station Below.
16. Airplane Observer Problem - Rate of Change of Angle of Elevation
17. Baseball Diamond Square Problem - Speed in ft/s.
18. Water trough problem