# Deflating balloon differentiation question

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#1
I am not sure whether the answer for this question is right:
A large spherical balloon is deflating.
At time t seconds the balloon has radius r cm and volume V cm3
The volume of the balloon is modelled as decreasing at a constant rate.
Given that
● the initial radius of the balloon is 40 cm
● after 5 seconds the radius of the balloon is 20 cm
● the volume of the balloon continues to decrease at a constant rate until the
balloon is empty

Q)solve the differential equation to find a complete equation linking r and t.
From this I calculated that dV/dr is 44800 pi/3
I then put them into the expression -k/r^2 =dr/dt where k is a positive constant, calculating k as -11200/3. From this I integrated by splitting variables and I get 1/3r^3 =11200/3 t +C. However, if I try and get a value for c (eg by substituting r=20 and t=5) I get -48000, which would not work if r was for example 40 (as this does not give t=0) So I am unsure where I have gone wrong.
Last edited by ahow39409234-095; 1 month ago
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1 month ago
#2
You forgot that dV/dr is negative (the balloon is decreasing in volume with time) so the coefficient in front of t term has a minus sign and C = 64,000. The balloon is fully deflated after approximately 5.714 seconds.
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#3
(Original post by lordaxil)
You forgot that dV/dr is negative (the balloon is decreasing in volume with time) so the coefficient in front of t term has a minus sign and C = 64,000. The balloon is fully deflated after approximately 5.714 seconds.
Thanks!
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1 month ago
#4
Hi, I'm currently doing the same question I believe. I am a little stuck, please could you help me? Part a asks you to show that dr/dt= -k/r^2 and I can't seem to get the answer. I'm probably being very stupid, but I used the formula for the volume of a sphere (v=4/3pir^3) to get dv/dr= 4Pir^2 and I also have dv/dt= -k (not sure about this bit). Then I used the chain rule to get dr/dt= dr/dv * dv/dt= 1/4pir^2 * -k which just gave me -k/4pir^2 and the 4pi bit shouldn't be there! I'm sure I've done something wrong, but I can't spot another way of doing it. Any advice would be greatly appreciated.
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1 month ago
#5
Hi, I'm currently doing the same question I believe. I am a little stuck, please could you help me? Part a asks you to show that dr/dt= -k/r^2 and I can't seem to get the answer. I'm probably being very stupid, but I used the formula for the volume of a sphere (v=4/3pir^3) to get dv/dr= 4Pir^2 and I also have dv/dt= -k (not sure about this bit). Then I used the chain rule to get dr/dt= dr/dv * dv/dt= 1/4pir^2 * -k which just gave me -k/4pir^2 and the 4pi bit shouldn't be there! I'm sure I've done something wrong, but I can't spot another way of doing it. Any advice would be greatly appreciated.
I haven't checked your general working, but k/4pi is just another constant, call it C, so you can write your answer as -C/r^2. It may just be a question of how you label your constants
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