Question

The rate of growth of the volume of a sphere is proportional to its volume. If the volume is initially 36pi ft^3 and expands to 90pi ft^3 after 1 second, find the V after 3 seconds.

Answer 1

calculus?

Answer 2

yes

Answer 3

\[\frac{ dV }{ dt } = kV\]
where k is the proportionality constant we need to find to solve this.

this can be re-arranged to:\[\frac{ dV }{ V } = kdt\]

if you integrate both sides, you get:\[\ln|V| + C_1 = kt + C_2\]which is equivalent to:\[\ln|V| = kt + C\]take e of both sides:\[e^{\ln|V|} = e^{kt + C}\]\[V = e^{kt}*e^C\]since e^C can be any constant, e^C = C. so we have:\[V = Ce^{kt}\]so, the volume is initially 36pi ft^3 (initially usually implies t = 0). so:\[36 \pi = Ce^{k*0}\] tells us:\[C = 36\pi\]so we plug the second one in to solve for k. at t = 1, V = 90:\[90\pi = 36\pi e^{k}\]\[\frac{ 90 }{ 36 } = e^k\]\[k = \ln \left( \frac{ 90 }{ 36 } \right)\]now we have k and C:\[V = 36\pi e^{0.916t}\]
Can you find V when t = 3?

Answer 4

at t = 1, V = 90pi***

Answer 5

V=1766
Thank you so much!

Answer 6

right answer. glad i could help C: