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i want to find V(t)

At first i found this problem was very simple but when i try to write differential equations i ended up with these

V' = kA thats for sure

then i confined the problem only to spherical shape and no other shapes of raindrops involved

as i cant express A in term of V alone( surface area of sphere = 4∏r

use chain rule, dV/dt= dV

V'=k4∏r

i get

4∏r

r'=k

r = kt+c

r

4∏r

im i correct? the answer to this problem is V'=kV

At first i found this problem was very simple but when i try to write differential equations i ended up with these

V' = kA thats for sure

then i confined the problem only to spherical shape and no other shapes of raindrops involved

as i cant express A in term of V alone( surface area of sphere = 4∏r

^{2}, volume of sphere is 4/3∏r^{3}) then i have touse chain rule, dV/dt= dV

_{dr}dr_{dt}substitute dV/dt fromV'=k4∏r

^{2}i get

4∏r

^{2}r'= K4∏r^{2}r'=k

r = kt+c

r

^{3}= (kt+c)^{3}4∏r

^{3}/3 = (kt+c)^{3}4∏/r=V(t)im i correct? the answer to this problem is V'=kV

^{2/3}im not sure how they transform Area to variable V alone
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