Question

the number n of locations of a popular coffeehouse chain is given in the table. (the numbers of locations as of october 1 are given.)

(a) find the average rate of growth between each pair of years.
2004 to 2006
locations/year
2006 to 2007
locations/year
2005 to 2006
locations/year

(b) estimate the instantaneous rate of growth in 2006 by taking the average of the last two rates of change in part (a).
locations/year

(c) estimate the instantaneous rate of growth in 2006 by measuring the slope of the secant line through (2005, 10243) and (2007, 15009).
locations/year

(d) estimate the instantaneous rate of growth in 2007 by measuring the slope of the secant line through (2006, 12439) and (2008, 16684).
locations/year

compare the growth rates you obtained in part (c) and (d). what can you conclude?
o there is not enough information.
o the rate of growth is decreasing.
o the rate of growth is constant.
o the rate of growth is increasing.

Explanation:

Step1: Calculate average rate for 2004 - 2006

The formula for average rate of change is $\frac{\Delta N}{\Delta t}=\frac{N_2 - N_1}{t_2 - t_1}$.
For 2004 - 2006: $t_1 = 2004$, $N_1=8565$, $t_2 = 2006$, $N_2 = 12439$.
$\frac{12439 - 8565}{2006 - 2004}=\frac{3874}{2}=1937$

Step2: Calculate average rate for 2006 - 2007

For 2006 - 2007: $t_1 = 2006$, $N_1 = 12439$, $t_2=2007$, $N_2 = 15009$.
$\frac{15009 - 12439}{2007 - 2006}=\frac{2570}{1}=2570$

Step3: Calculate average rate for 2005 - 2006

For 2005 - 2006: $t_1 = 2005$, $N_1 = 10243$, $t_2 = 2006$, $N_2 = 12439$.
$\frac{12439 - 10243}{2006 - 2005}=\frac{2196}{1}=2196$

Step4: Estimate instantaneous rate for 2006 (average of last two in (a))

Take average of 2005 - 2006 and 2006 - 2007 rates.
$\frac{2196+2570}{2}=\frac{4766}{2}=2383$

Step5: Estimate instantaneous rate for 2006 (slope through (2005,10243) and (2007,15009))

Using slope formula $\frac{y_2 - y_1}{x_2 - x_1}$.
$\frac{15009 - 10243}{2007 - 2005}=\frac{4766}{2}=2383$

Step6: Estimate instantaneous rate for 2007 (slope through (2006,12439) and (2008,16684))

$\frac{16684 - 12439}{2008 - 2006}=\frac{4245}{2}=2122.5$

Answer:

(a) 2004 to 2006: $1937$ locations/year; 2006 to 2007: $2570$ locations/year; 2005 to 2006: $2196$ locations/year
(b) $2383$ locations/year
(c) $2383$ locations/year
(d) $2122.5$ locations/year
Comparison conclusion: The rate of growth is decreasing.