the number n of locations of a popular coffeehouse chain is given in the table. (the numbers of locations as…

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.)\n\n(a) find the average rate of growth between each pair of years.\n2004 to 2006\nlocations/year\n2006 to 2007\nlocations/year\n2005 to 2006\nlocations/year\n\n(b) estimate the instantaneous rate of growth in 2006 by taking the average of the last two rates of change in part (a).\nlocations/year\n\n(c) estimate the instantaneous rate of growth in 2006 by measuring the slope of the secant line through (2005, 10243) and (2007, 15009).\nlocations/year\n\n(d) estimate the instantaneous rate of growth in 2007 by measuring the slope of the secant line through (2006, 12439) and (2008, 16684).\nlocations/year\n\ncompare the growth rates you obtained in part (c) and (d). what can you conclude?\no there is not enough information.\no the rate of growth is decreasing.\no the rate of growth is constant.\no the rate of growth is increasing.

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.)\n\n(a) find the average rate of growth between each pair of years.\n2004 to 2006\nlocations/year\n2006 to 2007\nlocations/year\n2005 to 2006\nlocations/year\n\n(b) estimate the instantaneous rate of growth in 2006 by taking the average of the last two rates of change in part (a).\nlocations/year\n\n(c) estimate the instantaneous rate of growth in 2006 by measuring the slope of the secant line through (2005, 10243) and (2007, 15009).\nlocations/year\n\n(d) estimate the instantaneous rate of growth in 2007 by measuring the slope of the secant line through (2006, 12439) and (2008, 16684).\nlocations/year\n\ncompare the growth rates you obtained in part (c) and (d). what can you conclude?\no there is not enough information.\no the rate of growth is decreasing.\no the rate of growth is constant.\no the rate of growth is increasing.

Answer

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.