Ch. 5 – Applications of Derivatives 5.6 – Related Rates

Ex: Find the derivative of the volume of a sphere with respect to its radius. Differentiate each variable with respect to r! Ex: Find the derivative of the volume of a sphere with respect to time. Now differentiate each variable with respect to t! This equation relates two rates, dV/dt and dr/dt, so this problem is a related rates problem! Related Rates problems are useful when observing the change in variables over time!

Ex: Find an equation that relates the volume of a cylinder over time with its radius and height over time. Use Product Rule!

Ex: A spherical kickball is being inflated. When the radius is 3 in Ex: A spherical kickball is being inflated. When the radius is 3 in., the volume is changing at a rate of 2π in3/s. What is the rate at which the radius is growing when the radius is 3 in? Always label what you know and what you need to find... ...then set up an equation that relates all of your rates. You’ll probably need to differentiate an existing equation with respect to t. Plug in the known values ONLY AFTER DIFFERENTIATING. Otherwise, you will be differentiating constants, not variables. Know Find

Ex: A man is 18 m away from a child Ex: A man is 18 m away from a child. The child releases a balloon that rises at a rate of 4 m/s. How quickly is the angle of elevation from the man to the balloon increasing 6 s after the child releases the balloon? Draw and label a picture to help you! Find an equation that relates your rates... At t = 6 , the height of the balloon will be... ...24 m, and using tanθ = 24/18, θ will be... ...≈.927 radians! Now plug n’ chug! M C B θ 18 m h Know Find