Respuesta :
Answer:[tex]\frac{32}{3}\left ( 1.0460\right )^t[/tex]
Step-by-step explanation:
Given
Population changes exponentially
[tex]P\left ( t\right )=ab^t[/tex]
[tex]P\left ( 9\right )=16[/tex]------1
[tex]P\left ( 9\right )=16=ab^8[/tex]
[tex]P\left ( 18\right =24=ab^{18}[/tex]-----2
divide 1 & 2 we get
[tex]\frac{24}{16}=\frac{b^{18}}{b^9}[/tex]
[tex]\frac{3}{2}=b^9 [/tex]
Substitute [tex]b^9[/tex] in 1 we get
[tex]a=\frac{32}{3}[/tex]
Thus [tex]P\left ( t\right )=\frac{32}{3}\left ( 1.0460\right )^t[/tex]
Answer:
[tex]P(t)=10.6667(1.0461)^t[/tex]
[tex]a\approx 10.6667[/tex]
[tex]b\approx 1.0461[/tex]
Step-by-step 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
[tex]P(t)=ab^t[/tex]
where, a is the initial population and b is growth factor.
P(9)=16 means P(t)=16 at x=9.
[tex]16=ab^9[/tex] .... (1)
P(18)=24 means P(t)=24 at x=18.
[tex]24=ab^{18}[/tex] .... (2)
Divide equation (2) by equation (1).
[tex]\dfrac{24}{16}=\dfrac{ab^{18}}{ab^9}[/tex]
[tex]1.5=b^9[/tex]
[tex](1.5)^{\frac{1}{9}}=b[/tex]
[tex]b\approx 1.0461[/tex]
Substitute [tex]b^9=1.5[/tex] in equation (1).
[tex]16=a(1.5)[/tex]
Divide both sides by 1.5.
[tex]\dfrac{16}{1.5}=a[/tex]
[tex]a\approx 10.6667[/tex]
Substitute the values of a and b in the given formula.
[tex]P(t)=10.6667(1.0461)^t[/tex]
Therefore, the formula for population is [tex]P(t)=10.6667(1.0461)^t[/tex].
