#### To determine

**To find:**

the time when the population was increasing most rapidly.

#### Answer

4.32 days

**Given: **The expression is p(t)=641+31e−0.7944t

**Formulae used:**

The derivative formulae

#### Explanation

Consider p(t)=641+31e−0.7944t

p(t)=641+31e−0.7944tp′(t)=64[24.6264e−0.7944t(1+31e−0.7944t)2]p″(t)=(1.5760896)[(−0.7944)e−0.7944t(1+31e−0.7944t)2−2(0−31×0.7944e−0.7944t)(1+31e−0.7944t)(1+31e−0.7944t)4]=(1.5760896)[49.2528e−0.7944t−0.7944e−0.7944t(1+31e−0.7944t)(1+31e−0.7944t)3]

Hence, use the formulae of derivative to find the p′(t) and p″(t)

To determine the extreme rate of population increase Let put p″(t)=0 to find the value of *x*

p″(t)=(1.576.0896)[49.2528e−0.7944t−0.7944e−0.7944t(1+31e−0.7944t)(1+31e−0.7944t)3]0=(1.576.0896)[49.2528e−0.7944t−0.7944e−0.7944t(1+31e−0.7944t)(1+31e−0.7944t)3]1+31e−0.7944t=49.2528e−0.7944t0.7944e−0.7944t1+31e−0.7944t=62

Further simplify

1+31e−0.7944t=62e−0.7944t=6131=1.9677t=4.32days

**Conclusion: **Hence the time when the population was increasing rapidly is 4.32 days.