Question

shown to the right is a certain population, in billions, for seven selected years from 1950 through 2007. using a graphing utilitys logistic regression option, we obtain the logistic growth model shown below for population, f(x), in billions, x years after 1949. how well does the function model the data for 2007?
$f(x)= \frac{10.54}{1 + 3.49e^{- 0.030x}}$

x, number of years after 1949	y, population (billions)
11 (1960)	2.9
21 (1970)	3.6
31 (1980)	4.4
41 (1990)	5.2
51 (2000)	6.0
58 (2007)	6.4

for 2007, the function dropdown the population to one decimal place.
options: slightly underestimates, accurately predicts, slightly overestimates

Explanation:

Step1: Determine x for 2007

2007 is 58 years after 1949, so \( x = 58 \).

Step2: Substitute x into the model

Substitute \( x = 58 \) into \( f(x)=\frac{10.54}{1 + 3.49e^{-0.030x}} \).
First, calculate the exponent: \( -0.030\times58=-1.74 \).
Then, calculate \( e^{-1.74}\approx0.175 \).
Next, calculate the denominator: \( 1 + 3.49\times0.175\approx1 + 0.611 = 1.611 \).
Finally, calculate \( f(58)=\frac{10.54}{1.611}\approx6.5 \).

Step3: Compare with actual data

The actual population in 2007 is 6.4 billion. The model's prediction (6.5) is slightly higher than 6.4.

Answer:

slightly overestimates