The number, N, of locations of a popular coffeehouse chain is given in the table
ID: 2836131 • Letter: T
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.)
Year 2004 2005 2006 2007 2008
N 8569 10241 12440 15011 16680
(a) Find the average rate of growth
( i ) from 2006 to 2008 ( ii ) from 2006 to 2007
( iii ) from 2005 to 2006
In each case, include the units.
(b) Estimate the instantaneous rate of growth in 2006 by
taking the average of two average rates of change. What
are its units?
(c) Estimate the instantaneous rate of growth in 2006 by measuring
the slope of a tangent.
(d) Estimate the intantaneous rate of growth in 2007 and compare
it with the growth rate in 2006. What do you conclude?
Explanation / Answer
There are two years that pass between 2006 and 2008
The value associated with 2006 is 12440
The value associated with 2008 is 16680
The rate of growth can be specified as either a percentage of the base amount (ie 12440) or the absolute change in value per year.
The absolute rate of change in value is found by subtracting the 2006 value from the 2008 value and dividing the result of the subtraction by two.
16680-12440=4240
4240/2=2120 items per year.
The percentage version of the rate of growth would be the absolute rate of change divided by the value of the variable associated with 2006.
2120/12440=17.04% per year.
Similar methods will allow you calculate the growth rates of the other two periods. (Each of the other periods is only one year in length.
b) The problem statement does not state when in the year the value of the variable is measured. If the data was measured on the first of July then the average rate of change from 2005 to 2007 would be a good estimate for the instantaneous rate of growth in 2006. The change between July first of 2005 to july first of 2007 would contain growth from 2006 plus half a years growth in 2005 and half a years growth in 2007.
Calculate the average for 2004 to 2008, 2005 to 2007, 2005 to 2006, 2006 to 2007
Let A48=average rate of change for 2004 to 2008
Let A57=average rate of change for 2005 to 2007
Let A56=average rate of change for 2005 to 2006
Let A67=average rate of change for 2006 to 2007
A good estimate for the instantaneous rate of change in 2006 would be
(A48 + 2*A57 +3*A56 +3*A67)/9
This methodology is called a weighted average.
C.
For the graph of data to have a slope at a particular data point requires one of two situations
one: The prior and following data points and the point in question are all on a single straight line.
two: Some software algorithm has smoothed the discrete data into a smooth curve.
The cord connecting the previous and next data point is a reasonably parallel to the tangent to most smooth curves that have been run through the data points.