# Mean Value problem

• Apr 14th 2007, 12:00 AM
RB06
Mean Value problem
Hi, I was wondering if anyone can help with this problem

Suppose that a tumor grows at the rate of r(t) = kt grams per week for some positive constant k, where t is the number of weeks since the tumor appeared. When, during the second 26 weeks of growth, is the mass of the tumor the same as its average mass during that period?

Any tips will be appreaciated.
• Apr 14th 2007, 01:07 AM
CaptainBlack
Quote:

Originally Posted by RB06
Hi, I was wondering if anyone can help with this problem

Suppose that a tumor grows at the rate of r(t) = kt grams per week for some positive constant k, where t is the number of weeks since the tumor appeared. When, during the second 26 weeks of growth, is the mass of the tumor the same as its average mass during that period?

Any tips will be appreaciated.

Let m be the tumor mass, then

dm/dt = k t,

so:

m= k t^2/2 + m0,

where m0 is the initial mass.

The avarage mass in the interval t1 to t2 is:

integral_{t=t1 to t2) m(t) dt / (t2-t1) = [k t2^3/6 + m0 t2 - kt1^3/6 - m0 t1]/(t2-t1)

So now we want to put t1=26, t2=52, and find t such that:

m(t) = [k 52^3/6 + m0 52 - k 26^3/6 - m0 26]/26 = k(2 52^2 - 26^2)/6 + m0

So:

t^2 = (2 52^2 - 26^2)3

hence:

t ~= 39.7

RonL

(check the arithmetic and algebra please).
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