To determine
To compute:
(a) limt→∞p(t)
(b) To find the rate of the spread of the tumor
Answer
(a) limt→∞p(t)=1
(b) The rate of the spread of the tumor is given by p′(t)=ake−kt(1+ae−kt)2
Explanation
(a) We know that as t→∞⇒e−∞=0
limt→∞p(t)=limt→∞11+ae−kt=1( since k>0)
Therefore as t→∞, the limt→∞P(t) is equal to 1.
(b) Given p(t)=11+ae−kt
Differentiating with respect to t, we get
p(t)=11+ae−ktWe know thatddx(uv)=vdu−udvv2p′(t)=ake−kt(1+ae−kt)2
Conclusion:
(a) limt→∞p(t)=1
(b) The rate of the spread of the tumor is given by p′(t)=ake−kt(1+ae−kt)2
a) $2: p(t)=11+ae−kt for the case a=10,k=0.5 and to find t such that 80% of the population hear the tumor

From the graph at t=7.38, we have p(t)=0.800179≈80%
Conclusion: t=7.38