Question

A population, P(t) (in millions) in year t, increases exponentially. Suppose P(9)=16 and P(18)=24. a) Find a formula for the population in the form P(t)=abt. Enter the values you found for a and b in your formula in the blanks below. Round your values to 4 decimal places.

Answer 1

`P(t)=10.6667(1.0461)^t`

`a\approx 10.6667`

`b\approx 1.0461`

Explanation:

It is given that a population, P(t) (in millions) in year t, increases exponentially.

The given values of he function are P(9)=16 and P(18)=24.

The formula for the population in the form

`P(t)=ab^t`

where, a is the initial population and b is growth factor.

P(9)=16 means P(t)=16 at x=9.

`16=ab^9` .... (1)

P(18)=24 means P(t)=24 at x=18.

`24=ab^(18)` .... (2)

Divide equation (2) by equation (1).

`(24)/(16)=(ab^(18))/(ab^9)`

`1.5=b^9`

`(1.5)^{(1)/(9)}=b`

`b\approx 1.0461`

Substitute
`b^9=1.5` in equation (1).

`16=a(1.5)`

Divide both sides by 1.5.

`(16)/(1.5)=a`

`a\approx 10.6667`

Substitute the values of a and b in the given formula.

`P(t)=10.6667(1.0461)^t`

Therefore, the formula for population is
`P(t)=10.6667(1.0461)^t`.

Answer 2

Answer:
`(32)/(3)\left ( 1.0460\right )^t`

Explanation:

Given

Population changes exponentially

`P\left ( t\right )=ab^t`

`P\left ( 9\right )=16`------1

`P\left ( 9\right )=16=ab^8`

`P\left ( 18\right =24=ab^(18)`-----2

divide 1 & 2 we get

`(24)/(16)=(b^(18))/(b^9)`

`(3)/(2)=b^9`

Substitute
`b^9` in 1 we get

`a=(32)/(3)`

Thus
`P\left ( t\right )=(32)/(3)\left ( 1.0460\right )^t`