Find the logistic function that satisfies the given conditions.
-Initial height $= 169$, limit to growth $= 845$, passing through $( 2,585 )$
A) $f ( x ) = \frac { 585 } { 1 + 4 \left( 1 / 9 ^ { x } \right. }$
B) $f ( x ) = \frac { 845 } { 1 + 2 ( 1 / 3 ) ^ { x } }$
C) $f ( x ) = \frac { 169 } { 1 + 3 ( 1 / 3 ) ^ { x } }$
D) $f ( x ) = \frac { 845 } { 1 + 4 ( 1 / 3 ) ^ { x } }$

Question

Quizplus · Accepted Answer

The general form of a logistic function is $f(x) = \frac{c}{1 + ae^{-bx}}$, where $c$ is the limit to growth. Here, $c = 845$. The initial height, when $x = 0$, is given by $f(0) = \frac{845}{1 + a} = 169$, which allows us to solve for $a$. The point $(2, 585)$ must satisfy the equation, which means when $x = 2$, $f(2) = 585$. Substituting the given values into the logistic function and solving for $a$ and $b$ (or the equivalent transformation for the logistic function), we find that option D, $f(x) = \frac{845}{1 + 4(1/3)^x}$, correctly models the situation with the given initial height, limit to growth, and passing through the given point.

Find the Logistic Function That Satisfies the Given Conditions =169= 169=169

Find the Logistic Function That Satisfies the Given Conditions $= 169$