Related Rates of Change

1. Related Rates of Change Problems where there are multiple variables & expressions. “Rate” refers to time unless otherwise stated. We link related rates using the chain rule

2. Approach • Identify what you are trying to find. • Draw pictures • Identify variables present in problem • Write chain rule so you end up with the desired rate. • Find equations to substitute into chain rule • Evaluate the derivative • Solve problem • Answer in context.

3. Eg: When a stone is dropped into a still pond of water, a circular ripple is formed. The radius of the circle increases at a rate of 2m/s. Calculate the rate at which the area of the circle is increasing when the radius is 8m. A circus strongman is inflating a spherical rubber hot-water bottle at a steady rate of 1280cm3/s (V=(4/3)πr3) Calculate the rate at which the radius is increasing at a time 1min after the strongman has started inflating the bottle.

4. Eg: When a stone is dropped into a still pond of water, a circular ripple is formed. The radius of the circle increases at a rate of 2m/s. Calculate the rate at which the area of the circle is increasing when the radius is 8m. • We are trying to find dA/dt when r=8m • Varibles we have are r, A, t • A=πr2 dA/dr=2πr • dr/dt=2 • When r=8, dA/dt=32π=100.5m2/s

5. A circus strongman is inflating a spherical rubber hot-water bottle at a steady rate of 1280cm3/s (V=(4/3)πr3) Calculate the rate at which the radius is increasing at a time 1min after the strongman has started inflating the bottle. • We are trying to find dr/dt when t=60 • Varibles: r, V, t • dV/dt=1280 • dV/dr=4 πr2 • dr/dt=1280/4 πr2 • When t=60, dr/dt=??? • Find r when t=60 (V increases by 1280cm3/s : after 60 sec, V=76800, use V=(4/3)πr3 to find r=26.4cm • Substitute 4=26.4 into dr/dt=1280/4 πr2 • dr/dt=0.146cm/s after 60 seconds