#### 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