Let g(t) be the Glucose concentration in blood stream of a person at time t.
g(t) = 1/k(R - (R - k*go)e^(-kt/v))
Where go - initial glucose concentration (at t=0),
v - volume of blood in Humana body ,
R - rate at which glucose is introduced into blood stream and
k is he fraction of glucose removed from blood stream
Find an expression for glucose concentration in blood stream of a person after a long time.
So here when t tends to infinity the terms inside the second bracket is divided by a very large value and does they tends to zero or is there any effect of k , even t tends to infinity?
e^- (k oo) -----> zero
no matter how big k is (short of oo :)
That means g(t) tends to 1/k(R) right?
To find the glucose concentration in the blood stream after a long time (as t tends to infinity), we can analyze the expression:
g(t) = 1/k(R - (R - k*go)e^(-kt/v))
As t tends to infinity, the term e^(-kt/v) will tend to zero. This is because for any positive constant k, the exponential function e^(-kt/v) approaches zero as t becomes very large.
Therefore, we can simplify the expression for g(t) as t tends to infinity:
g(t) = 1/k(R - (R - k*go)e^(-kt/v))
= 1/k(R - (R - k*go)*0) (since e^(-kt/v) = 0 as t tends to infinity)
= 1/k(R - 0)
= 1/k(R)
= R/k
Thus, the glucose concentration in the blood stream of a person after a long time (as t tends to infinity) is R/k. So, the value of k does have an effect on the glucose concentration even as t tends to infinity.