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