Question

The volume of spherical balloon been intimated synthes a constant rate if initial its radius is 3 unit and after 3 second it is 600 find the radii of the balloon after t seconds

Answer 1

✨Here is your answer
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Step 1:

Volume of the balloon V=43V=43πr3πr3

Diff w.r.t tt we get,

dvdt=43dvdt=43π3r2drdtπ3r2drdt

When t=0,r=t=0,r=3 units

When t=3,r=t=3,r=6 units

It is given dvdtdvdt=constant=k(say)

∴k=4πr2drdt∴k=4πr2drdt

Separating the variables we get,

∴kdt=4πr2dr



Step 2:

Integrate on both sides

k∫dt=4π∫r2dr

kt=4πr33+c------(1)

When t=0,r=3units

⇒k(0)=4π(3)33+c

c=36π

Step 3:

When t=3sec,r=6units

⇒k(3)=4π(6)33+36π

3k=288π+36π

k=324π3

=108π

∴ Hence c=36π and k=108π

Step 4: see the picture.
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