<p>hi all
i have a question i can't really get so i'd appreciate any help you have to give, all you math geniuses you.
(isnt flattery always the way to go?)
at time t, t> or = to 0, the volume of a sphere is increasing at a rate proportional to the reciprocal of its radius. at t=0, the radius of the sphere is 1 and at t=15, the radius is 2. (the vol v os a sphere w. radius r is v=4/3pi r^3)
a. find the radius of the sphere as a function of t
b. at what itme t will the vol of the sphere be 27 times its vol at t=0?</p>
<p>so i know how to find dv/dt but idk how to get r in terms of t only. any direction would help
thanks!</p>
<p>you know you want to helppp</p>
<p>well i'm just going to tell u the basics. i feel a little lazy to do the done thing but i will help u provided u give me the calculations.</p>
<p>a. dV/dt=k/r -> dV/dr*dr/dt=k/r
change this to dr/dt=
pluge in r=1 at t=0 and r=2 at t=15 to figure out k and stuff.
then u just set the equation so that v at t=x is 27 times that of V at t=0</p>
<p>which part?... actually this is really not easy to tell u on collegeconfidential. so im me or sth</p>
<p>I'm not sure if this is right becuase it doesn't quite write r as a function of t, but here is how I would do the problem:</p>
<p>a.
dV/dt = k/r where k is an known constant of proportionality
dV = k/r dt<br>
V = t<em>k/r + C where C is an inknown constant of integration
4/3</em>pi<em>r^3 = t</em>k/r + C</p>
<p>plug in t=0 and r=1 to find that C = r/3<em>pi
plug in t=15 and r=2 and C=r/3</em>pi to find k= 56pi/45</p>
<p>now our equation is 4/3<em>pi</em>r^3 = t<em>(56pi/45) / r + 4/3pi
multiply both sides by r</em>3/(4pi) to get
r^4=t*14/15+r
I don't think its possible to write r as a function of t, but given a t value you could use a calculator to solve for r.</p>
<p>b. When t=0, r=1 so V=(4/3)pi<em>1^3 = (4/3)pi
27 * (4/3)pi = 36 pi = our deired volume
Find our desired r:
36pi = (4/3)pi</em>r^3
27=r^3
3=r</p>
<p>Use our equation from a:
3^4=t<em>14/15+3
81=t</em>14/15+3
78=t*14/15
585/7=t</p>
<p>If anyone finds the right answer, post it here; I want to know what I'm missing in part a.</p>
<p>a
r = at + b where a and b are constants
1 = a(0) + b
2 = a(15) + b
b = 1
a = (1/15)
r = (1/15)t + 1</p>
<p>b
V = (4/3)(pi)(r^3) = (4/3)(pi)[(1/15)t + 1]^3
V(0) = (4/3)(pi)[(1/15)t + 1]^3 = (4/3)(pi)[(1/15)(0) + 1]^3 = (4/3)(pi)
V(t) = (4/3)(pi)[(1/15)t + 1]^3
(4/3)(pi) * 27 = (4/3)(pi)[(1/15)t + 1]^3
27 = [(1/15)t + 1]^3
3 = (1/15)t + 1
(1/15)t = 2
t = 30
At t = 30, the volume is 27 times larger than when t = 0.</p>
<p>that was the non calculus method. here's the method w/ calculus.</p>
<p>V = (4/3)(pi)(r^3)
dV/dr = 4(pi)(r^2)
dV/dt = k/r
k is a constant</p>
<p>a)
dr/dt = (dr/dV) * (dV/dt)
dr/dt = {1 / [4(pi)(r^2)]} * (k/r) = k/[4(pi)(r^3)]
<a href="dr">4(pi)(r^3)</a> = k(dt)
(pi)(r^4) = kt + c
c is a constant</p>
<p>r(0) = [c/(pi)]^(1/4) = 1; c = pi
r(15) = [(15k + pi)/(pi)]^(1/4) = 2; k = pi
r(t) = (t + 1)^(1/4)</p>
<p>b)
dV/dt = (pi)/r
dV/dt = (pi) / [(t + 1)^(1/4)] = (pi)[(t + 1)^(-1/4)]
V(t) = (pi)(4/3)[(t + 1)^(3/4)]
V(0) = (pi)(4/3)
V(t) / V(0) = 27
t = 80</p>