Section 4.4 Related Rates (Minimum Homework 1, 3, 5, 7)

Section 4.4 Related Rates (Minimum Homework: 1, 3, 5, 7) 1) A pebble is dropped into a calm pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area of the disturbed water changing? 1. Identify the information that is given. Let 𝑟 = 𝑜𝑢𝑡𝑒𝑟 𝑟𝑎𝑑𝑖𝑢𝑠 𝑑𝑟 𝑑𝑡 = 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 𝑖𝑠 𝑔𝑟𝑜𝑤𝑖𝑛𝑔 𝑖𝑛 𝑓𝑒𝑒𝑡 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑 Given: 𝑑𝑟 = 1 𝑓𝑜𝑜𝑡 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑡 𝑟 = 4 𝑓𝑡 2. Identify what needs to be solved for. 𝑊𝑒 𝑎𝑟𝑒 𝑎𝑠𝑘𝑒𝑑 𝑡𝑜 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑟𝑎𝑡𝑒 𝑎𝑡 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑠𝑡𝑢𝑟𝑏𝑒𝑑 𝑤𝑎𝑡𝑒𝑟 𝑖𝑠 𝑐ℎ𝑎𝑛𝑔𝑖𝑛𝑔. Symbolically, we are asked to find 𝑑𝐴 𝑑𝑡 where 𝐴 represents the disturbed area. 3. Write an equation that relates the variables. To derive the equation you may use a geometric fact (like an area or a volume formula) 𝑊𝑒 𝑛𝑒𝑒𝑑 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝑓𝑜𝑟 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒: 𝐴 = 𝜋𝑟 2 𝑑 4. Take the derivative 𝑑𝑡 of both sides of the equation. 𝑑 𝑑𝑡 (𝐴) = 𝑑𝐴 𝑑 𝑑𝑡 (𝜋𝑟 2 ) 𝑑𝑟 1 𝑑𝑡 = 2𝜋𝑟 𝑑𝑡 5. Solve for the unknown rate of change. 𝑑𝐴 𝑑𝑡 𝑑𝑟 = 2𝜋𝑟 𝑑𝑡 6. Substitute all known values to get the final answer. 𝑑𝑟 Substitute 𝑟 = 4 𝑎𝑛𝑑 𝑑𝑡 = 1 𝑑𝐴 𝑑𝑡 𝑑𝐴 𝑑𝑡 = 2𝜋(4)(1) = 8𝜋 Now write an answer with units. Answer: The disturbed area is growing at a rate of 8𝜋 𝑓𝑒𝑒𝑡 𝑠𝑞𝑢𝑎𝑟𝑒𝑑 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑 3) A pebble is dropped into a calm pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 3 feet per second. When the radius is 5 feet, at what rate is the total area of the disturbed water changing? 1. Identify the information that is given. Let 𝑟 = 𝑜𝑢𝑡𝑒𝑟 𝑟𝑎𝑑𝑖𝑢𝑠 𝑑𝑟 𝑑𝑡 = 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 𝑖𝑠 𝑔𝑟𝑜𝑤𝑖𝑛𝑔 𝑖𝑛 𝑓𝑒𝑒𝑡 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑 Given: 𝑑𝑟 = 3 𝑓𝑒𝑒𝑡 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑡 𝑟 = 5 𝑓𝑡 2. Identify what needs to be solved for. 𝑊𝑒 𝑎𝑟𝑒 𝑎𝑠𝑘𝑒𝑑 𝑡𝑜 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑟𝑎𝑡𝑒 𝑎𝑡 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑠𝑡𝑢𝑟𝑏𝑒𝑑 𝑤𝑎𝑡𝑒𝑟 𝑖𝑠 𝑐ℎ𝑎𝑛𝑔𝑖𝑛𝑔. Symbolically, we are asked to find 𝑑𝐴 𝑑𝑡 where 𝐴 represents the disturbed area. 3. Write an equation that relates the variables. To derive the equation you may use a geometric fact (like an area or a volume formula) 𝑊𝑒 𝑛𝑒𝑒𝑑 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝑓𝑜𝑟 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒: 𝐴 = 𝜋𝑟 2 𝑑 4. Take the derivative 𝑑𝑡 of both sides of the equation. 𝑑 𝑑𝑡 (𝐴) = 𝑑𝐴 𝑑 𝑑𝑡 (𝜋𝑟 2 ) 𝑑𝑟 1 𝑑𝑡 = 2𝜋𝑟 𝑑𝑡 5. Solve for the unknown rate of change. 𝑑𝐴 𝑑𝑡 𝑑𝑟 = 2𝜋𝑟 𝑑𝑡 6. Substitute all known values to get the final answer. 𝑑𝑟 Substitute 𝑟 = 5 𝑎𝑛𝑑 𝑑𝑡 = 3 𝑑𝐴 𝑑𝑡 𝑑𝐴 𝑑𝑡 = 2𝜋(5)(3) = 30𝜋 Now write an answer with units. Answer: The disturbed area is growing at a rate of 30𝜋 𝑓𝑒𝑒𝑡 𝑠𝑞𝑢𝑎𝑟𝑒𝑑 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑 5) Air is being pumped into a spherical balloon at 10 cm3/minute. Calculate the rate at which the radius of the balloon is changing when the radius of the balloon is 6 cm. 1. Identify the information that is given. Let 𝑟 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑏𝑎𝑙𝑙𝑜𝑜𝑛 𝑉 = 𝑣𝑜𝑙𝑢𝑚𝑒 (𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑎𝑖𝑟 𝑖𝑛𝑠𝑖𝑑𝑒 𝑡ℎ𝑒 𝑏𝑎𝑙𝑜𝑜𝑛) Given: 𝑟 = 6 𝑐𝑚 Given: 𝑑𝑉 𝑑𝑡 = 10 𝑐𝑚3 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑 2. Identify what needs to be solved for. We are asked to find how fast the radius is changing. 𝑑𝑟 That is we are asked to find 𝑑𝑡 (hint units of our answer will be cm/sec as the length of the radius is given in centimeters, and time is given in seconds) 3. Write an equation that relates the variables. To derive the equation you may use a geometric fact (like an area or a volume formula) We need the volume formula for a sphere. 4 𝑉 = 3 𝜋𝑟 3 𝑑 4. Take the derivative 𝑑𝑡 of both sides of the equation. 𝑑 𝑑𝑡 (𝑉) = 𝑑 4 ( 𝜋𝑟^3) 𝑑𝑡 3 𝑑𝑉 4 𝑑𝑟 1 ∗ 𝑑𝑡 = (3 ∗ 3 𝜋𝑟 2 ) 𝑑𝑡 𝑑𝑉 𝑑𝑡 𝑑𝑟 = 4𝜋𝑟 2 𝑑𝑡 5. Solve for the unknown rate of change. 𝑤𝑒 𝑛𝑒𝑒𝑑 𝑡𝑜 𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑑𝑉 𝑑𝑡 = 𝑑𝑟 4𝜋𝑟 2 𝑑𝑡 Divide both sides by 4𝜋𝑟 2 𝑑𝑟 𝑑𝑡 𝑑𝑉 = 𝑑𝑟/𝑑𝑡 𝑑𝑡 4𝜋𝑟2 𝑑𝑟 𝑑𝑡 = 𝑑𝑉 𝑑𝑡 4𝜋𝑟2 6. Substitute all known values to get the final answer. Let 𝑑𝑟 𝑑𝑡 𝑑𝑉 𝑑𝑡 = 10 𝑎𝑛𝑑 𝑟 = 6 10 10 5 = 4𝜋(6)2 = 144𝜋 = 72𝜋 The units in my answer should be centimeters per second, as the radius is measured in centimeters and the time is given in seconds. 5 Answer: Radius is growing at a rate of 72𝜋 𝑐𝑒𝑛𝑡𝑖𝑚𝑒𝑡𝑒𝑟𝑠 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑 7) Air is being pumped into a spherical balloon at 3 cm3/minute. Calculate the rate at which the radius of the balloon is changing when the radius of the balloon is 8 cm. 1. Identify the information that is given. Let 𝑟 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑏𝑎𝑙𝑙𝑜𝑜𝑛 𝑉 = 𝑣𝑜𝑙𝑢𝑚𝑒 (𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑎𝑖𝑟 𝑖𝑛𝑠𝑖𝑑𝑒 𝑡ℎ𝑒 𝑏𝑎𝑙𝑜𝑜𝑛) Given: 𝑟 = 8 𝑐𝑚 Given: 𝑑𝑉 𝑑𝑡 = 3 𝑐𝑚3 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑 2. Identify what needs to be solved for. We are asked to find how fast the radius is changing. 𝑑𝑟 That is we are asked to find 𝑑𝑡 (hint units of our answer will be cm/sec as the length of the radius is given in centimeters, and time is given in seconds) 3. Write an equation that relates the variables. To derive the equation you may use a geometric fact (like an area or a volume formula) We need the volume formula for a sphere. 4 𝑉 = 3 𝜋𝑟 3 𝑑 4. Take the derivative 𝑑𝑡 of both sides of the equation. 𝑑 𝑑𝑡 (𝑉) = 𝑑 4 ( 𝜋𝑟^3) 𝑑𝑡 3 𝑑𝑉 4 𝑑𝑟 1 ∗ 𝑑𝑡 = (3 ∗ 3 𝜋𝑟 2 ) 𝑑𝑡 𝑑𝑉 𝑑𝑡 𝑑𝑟 = 4𝜋𝑟 2 𝑑𝑡 5. Solve for the unknown rate of change. 𝑤𝑒 𝑛𝑒𝑒𝑑 𝑡𝑜 𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑑𝑉 𝑑𝑡 = 𝑑𝑟 4𝜋𝑟 2 𝑑𝑡 Divide both sides by 4𝜋𝑟 2 𝑑𝑟 𝑑𝑡 𝑑𝑉 = 𝑑𝑟/𝑑𝑡 𝑑𝑡 4𝜋𝑟2 𝑑𝑟 𝑑𝑡 = 𝑑𝑉 𝑑𝑡 4𝜋𝑟2 6. Substitute all known values to get the final answer. Let 𝑑𝑟 𝑑𝑡 𝑑𝑉 𝑑𝑡 = 3 𝑎𝑛𝑑 𝑟 = 8 3 3 3 = 4𝜋(8)2 = 4∗64𝜋 = 256𝜋 The units in my answer should be centimeters per second, as the radius is measured in centimeters and the time is given in seconds. 3 Answer: Radius is growing at a rate of 256𝜋 𝑐𝑒𝑛𝑡𝑖𝑚𝑒𝑡𝑒𝑟𝑠 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑