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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.

2 Answers

3 votes

Answer:


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.

User Adeeb
by
6.9k points
3 votes

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

User Arun Pratap Singh
by
7.4k points