What Is the Rate of Change of the Cone's Height When the Radius Is 3 Units?

[NewsboysGurl91]

The volume V of a cone (V = 1/3*pi*r^2*h) is increasing at the rate of 28pi cubic units per second. At the instant when the radius r of the cone is 3 units, its volume is 12pi cubic units and the radius is increasing at 1/2 unit per second. At the instant when the radius of the cone is 3 units, what is the rate of change of the area of its height h?

Know: dV/dt = 28pi cubic units per second, dr/dt = 1/2 unit per second, and dA/dt = 3pi units square per second. Need: dh/dt
12pi = 1/3pi*3^2*h
h=4

V = 1/3*pi*r^2*h
dV/dt = 1/3*pi*2r*dr/dt*h + r^2*dh/dt (use the product rule. do you multiply 1/3 pi times everything like this 1/3*pi(2r*dr/dt*h + r^2*dh/dt) or how I did it?)
28pi = 1/3pi*2(3)*1/2*4 + 6*dh/dt
24pi = 6*dh/dt
dh/dt = 4pi units per second



Is this correct?

 

[HallsofIvy]

The volume V of a cone (V = 1/3*pi*r^2*h) is increasing at the rate of 28pi cubic units per second. At the instant when the radius r of the cone is 3 units, its volume is 12pi cubic units and the radius is increasing at 1/2 unit per second. At the instant when the radius of the cone is 3 units, what is the rate of change of the area of its height h?

Know: dV/dt = 28pi cubic units per second, dr/dt = 1/2 unit per second, and dA/dt = 3pi units square per second. Need: dh/dt
12pi = 1/3pi*3^2*h
h=4

V = 1/3*pi*r^2*h
dV/dt = 1/3*pi*2r*dr/dt*h + r^2*dh/dt (use the product rule. do you multiply 1/3 pi times everything like this 1/3*pi(2r*dr/dt*h + r^2*dh/dt) or how I did it?)

What you have is wrong. You can either think of this as V= (1/3)(pi r^2h) so that dV/dt= (1/3)d(pi r^2h)/dt or do it as V= ((1/3)pir^2)(h) so that dV/dt= {d((1/3)pir^2)/dt}h+ ((1/3)pi r^2)(dh/dt). Either way, the (1/3)pi multiplies the entire derivative:
dV/dt= (2/3)pi rh dr/dt+ (1/3)pi r^2 dh/dt.

28pi = 1/3pi*2(3)*1/2*4 + 6*dh/dt

28pi= (1/3)pi(2)(3)(1/2)(4)+ (2/3)pi (9) dh/dt
(Where did you get that "6"? if r= 3, r^2= 9, not 18!

24pi = 6*dh/dt
dh/dt = 4pi units per second



Is this correct?

No, it isn't!