Related rates:Let V be the volume of a cylinder having height h and radius r, and assume that h and r vary with time

Question

Let V be the volume of a cylinder having height h and radius r, and assume that h and r vary with time

a.) how are dV/dt, dh/dt, and dr/dt related?

dv/dt= f(r,h)dh/dt+g(r,h)=dr/dt, where f(r,h)=? and g(r,h)=?

 

I have no idea how to figure out what f(r,h)=? and g(r,h)=? are equal to. 

 

I know V=(pi)r^2h 

and dv/dt= (pi)(r^2*dh/dt +2rh*dr/dt)

 

But how can i figure out the f(r,h) and g(r,h)

 

b.)At a certain instant, the height is 8 in and increasing at 1 in/s, while the radius is 11 in and decreasing at 1 in/s. How fast is the volume changing at that instant? Is the volume increasing or decreasing at that instant?

 

find dv/dt| r=11,h=8 given that dh/dt|r=11,h=8 =1 and dr/dt|r=11,h=8 =-1

 

I found that dv/dt=-55(pi)in^3/s and the volume is decreasing

*I really just need help with part a!!

Answer 1

You are on the right track and just need to see what is exactly requested.

 

V = πr²h

 

dV/dt = 2πrh dr/dt + πr² dh/dt

 

dV/dt = g(r,h) dr/dt + f(r,h) dh/dt

 

Just compare the dr/dt parts and the dh/dt parts for the last two equations to figure out f(r,h) and g(r,h).