1. 3.9 Related Rates 1

2. Example Assume that oil spilled from a ruptured tanker in a circular pattern whose radius increases at a constant rate of 2 ft/s. How fast is the area of the spill increasing when the radius of the spill is 60 ft? This is a related rate problem. The rate the area is increasing is related to the rate the radius is increasing. By what formula?

3. Guidelines 1.Read the problem and understand what is being asked. 2.Draw a diagram and assign variables for the situation. (Use d/dt for quantities that change) 3.Write an equation or formula that relates the quantities in the situation. 4.Use implicit differentiation and differentiate each variable with respect to the third variable, usually time. (Don’t forget Chain Rule) 5. Substitute the given information into the resulting equation. 6.Solve for the unknown rate.

4. If the radius of a sphere is changing at an instantaneous rate of 0.1 cm/sec, how fast is its volume changing when the radius is 10 cm? The sphere is growing at a rate of. Sphere Problem Given: Find:

5. Hot Air Balloon Problem Given: A balloon rises from a point on the ground 500ft from an observer. If the angle of elevation is increasing at a rate of 0.14 rad/min. How fast is the balloon rising when its angle of elevation is 45 degree ? Find:

6. Example http://astro.temple.edu/~dhill001/relatedrates/shadowdemo.gif If the ladder is falling down at a rate of 8 ft/s, how fast is the bottom of the ladder sliding away when it is 6 feet from the wall?

7. B A Truck Problem Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr. How fast is the distance between the trucks changing 6 minutes later?

8. B A Truck Problem How fast is the distance between the trucks changing 6 minutes later? Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr.

9. Example http://astro.temple.edu/~dhill001/relatedrates/shadowdemo.gif A man 6 ft tall is directly under a light which is 16 ft high. If he walks to the right at a steady speed of 4 ft/sec, how fast is the length of his shadow increasing when he is 5 feet away from the light post?

10. Example

11. 1. A base runner is running at the speed of 10 ft/sec from first base to second base. How fast is his distance from home changing when he is halfway between the first and second base? 2.If a snowball melts so that its surface area decreases at a rate of 2 cm/min, find the rate at which the diameter decreases when the diameter is 10cm. Group work