A spherical balloon bursts when the surface area(S) exceeds 500 cm^2. If the radius changes at a rate of 23 cm/s, what is the rate of change of volume at the instant the balloon bursts?

(V=(4/3)(pi)r^3) (S=4(pi)r^2)

Question

A spherical balloon bursts when the surface area(S) exceeds 500 cm^2. If the radius changes at a rate of 23 cm/s, what is the rate of change of volume at the instant the balloon bursts?

(V=(4/3)(pi)r^3) (S=4(pi)r^2)

anonymous · Answer

i would set the surface area equal to 500 cm^2, to find out what the radius is when the balloon will burst:
$$500 = 4\pi r^2 \Rightarrow r = \sqrt{\frac{500}{4\pi}}$$

anonymous · Answer

r=$$\sqrt{125\pi}$$ when balloon burst..

anonymous · Answer

Then, take the derivative of the Volume equation, and plug in that value of r and dr/dt (which is 23 cm/s)

amistre64 · Answer

V=(4/3)(pi)r^3  ; derive the sides

dV/dt = (2)(4pi/3) * r^2 * dr/dt ; this gives us the rate of change of the volume with respect to time

amistre64 · Answer

since dr/dt = 23 as stated in the problem; all we have to do is determine the value of "r" to solve right?

amistre64 · Answer

and of course we have to correct for my inherent stupidity :)

dV/dt = (3)(4pi/3) * r^2 * dr/dt ; changed the 2 to 3 lol

anonymous · Answer

So... What you're saying is...?

amistre64 · Answer

it might be interesting to note that the derivative of volume becomes surface area ....

amistre64 · Answer

dV/dt = (3)(4pi/3) * r^2 * dr/dt

V' = 4pi r^2 * dr/dt ; and  dr/dt = 23 so
    = 4pi r^2 * 23

anonymous · Answer

F'(x)=4/3pir^3  ?

anonymous · Answer

12/9 pi r^2 ?

amistre64 · Answer

since r = sqrt(125/pi) when it bursticates

V' = 4pi (sqrt(125/pi))^2 * 23
    = 4pi (125/pi)(23)             ; the pis cancels
    = 4(125)(23)                    ; and whatever that equals in the end

anonymous · Answer

dv/dt=d(4/3*pi*r^3)/dt=4/3*3*pi*r^2*(dr/dt) =4pi*r^2*(dr/dt)=500*23
The key is the chain rule. I think

amistre64 · Answer

no, there is no chain rule needed in this one

anonymous · Answer

ohhhh!!! I see. kind of :x Lol. What do you mean by bursticates? ami?(:

amistre64 · Answer

Kr^3 derives to 3Kr^2, no chains required :)

amistre64 · Answer

bursticate, when the bubble pops; like when the housing market took a dive ;)

anonymous · Answer

ohhhh! okay thankss!

amistre64 · Answer

i think i see what prot was saying; the dr/dt pops out in the end ... yes

amistre64 · Answer

i dont think that is the chain rule perse; Ive always called it a "derived bit"

amistre64 · Answer

its always there, but when its like dx/dx it just goes to 1 and vanishes from teh scene

anonymous · Answer

Anyway. dv/dt=(dv/dr)*(dr/dt) 
V=4/3pi*r^3 
I think it is the chain rule. But That's ok