Lecture 14 related rates - section 4.1
- 2. 2 Example 1 – Inflating a Balloon Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm? Solution: We start by identifying two things: the given information: the rate of increase of the volume of air is 100 cm3/s and the unknown: the rate of increase of the radius when the diameter is 50 cm
- 3. cont’d 3 Example 1 – Solution In order to express these quantities mathematically, we introduce some suggestive notation: Let V be the volume of the balloon and let r be its radius. Key idea: rates of change are derivatives The rate of increase of the volume with respect to time is the derivative dV/dt, the rate of increase of the radius is dr /dt.
- 4. 4 Example 1 – Solution Given: = 100 cm3/s Unknown: when r = 25 cm connect dV/dt and dr/dt relate V and r V = pr 3 cont’d
- 5. 5 Example 1 – Solution Differentiate each side of this equation with respect to t. For right side, we need to use the Chain Rule: Now we solve for the unknown quantity: cont’d
- 6. 6 Example 1 – Solution If we put r = 25 and dV/dt = 100 in this equation, we obtain The radius of the balloon is increasing at the rate of 1/(25p) ≈ 0.0127 cm/s. cont’d
- 7. 7 Solving Related Rates - Process 1. Read the problem (several times) 2. Draw a diagram 3. Label variables 4. Identify givens 5. Identify what you are trying to find (usually a derivative) 6. Relate variables equation May need to eliminate varaibles 1. Differentiate with respect to time 2. Solve for unknown 3. Answer question, with units!!
- 8. Example – Eliminating a Variable A water tank has the shape of an inverted circular cone with a base radius of 2 m and height of 4 m. If water is being pumped into the tank at a rate of 2 m3/min, find the rate at which the water level is rising when the water is 3 m deep. 8
- 9. Example – Implicit Differentiation A ladder 10 ft 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 ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall? 9
- 10. 10 Example A man walks along a straight path at a speed of 4 ft/s. A searchlight is located on the ground 20 ft from the path and is kept focused on the man. At what rate is the seachlight rotating when the man is 15 ft from the point on the path closest to the searchlight?