discrete calc problem with spheres

Question

A sphere is growing at a rate proportional to the radius. At t=0 the radius is 8in and at t=30 the radius is 12in. What is the radius at t=15?

 

I'm stumped on this one. I've worked with taking derivatives of the volume and trying to solve for a constant k related to the radius, but I don't think I'm on the right track.

Answer 1

Hi Katie 

        Here's my take if r= radius then you were given dr/dt=kr solving this gives r(t)=e^kt+c where c and k are constants to be determined. at t=0 r=8"  so 8=e^c so c=ln(8)= 2.07 and t=30 r=12" so 12=e^30k+2.07 so ln12 =30k +2.07, solving for k gives k=(ln(3/2)/30=.0135

 

The radius at t=15 r(15)= 9.71" is obtained from r(t)=e^.0135*t+2.07

 

Hope this helps

Jim