#### To determine

**(**a)

**To find:** The number of bacteria after *t* hours.

**Solution: **y=200(3.24)t.

**Given:** The initial amount a=200 cells

**Formulae used:** The standard exponential growth function

#### Explanation

Consider y=a(b)t

Hence, use the formulae of standard exponential growth function

Now according to the given expression, the value of the given expression is

y=a(b)t=200(b)t

Use the given initial condition, after half an hour the population has increased to *360* cells

Therefore,

y=200(b)t360=200(b)12b12=1.8b=3.24

Thus, the general expression is y=200(3.24)t

**Conclusion:** Hence the general expression is y=200(3.24)t

**(b)**

**To find:** The number of bacteria after *4* hours.

**Solution: **≈22040cells.

**Given:** The time, t=4 hours

**Formulae used:** The general expression of growth of bacteria is y=200(3.24)t

Consider y=200(3.24)t

Hence, use the formulae of general expression of growth of bacteria is y=200(3.24)t

Now according to the given expression, the value of the given expression when t=4 is

y=200(3.24)t=200(3.24)4≈22040cells

**Conclusion:** Hence the number of bacteria after *4* hours is ≈22040cells

**(c)**

**To find:** The rate of bacteria after *4* hours.

**Solution: **≈25910bacteria/hours.

**Given:** The time, t=4 hours

**Formulae used:** The derivative of the expression with respect to time

Consider y=200(3.24)t

Hence, use the formulae of derivative of function

Now according to the given expression, the value of rate of change when t=4 is

y=200(3.24)ty′=200(3.24)tlog3.24y′=200log3.24(3.24)4≈25910bacteria/hours

**Conclusion:** Hence the rate of change of bacteria after *4* hours is ≈25910bacteria/hours

**(d)**

**To find:** The time for which the population reach *10000*.

**Solution: **t≈3.3hr.

**Given:** The population, y=10000

**Formulae used:** The general expression of growth of bacteria is y=200(3.24)t

Consider y=200(3.24)t

Hence, use the formulae of general expression of growth of bacteria is y=200(3.24)t

Now according to the given expression, the value of rate of change when t=4 is

y=200(3.24)t10000=200(3.24)t(3.24)t=50t≈3.3hr

**Conclusion:** Hence the value of time for which population reach 10000 is t≈3.3hr.