c) Estimate the instantaneous rate of growth in 2006 by measuring the slope of the tangent line through
(2005, 10237) and (2007, 15005).
d) Estimate the instantaneous rate of growth in 2007 by measuring the slope of the tangent line through
(2006, 12435) and (2008, 16684).
e)Compare the growth rates you obtained in part (c) and (d). What can you conclude?
c) Well, to estimate the instantaneous rate of growth in 2006, we need to measure the slope of the tangent line. Let's see... hmm, it seems like we have the coordinates (2005, 10237) and (2007, 15005). Okay, here we go. *puts on imaginary math glasses* Ahem. So, we need to calculate the slope of the line between those two points.
Slope = (y2 - y1) / (x2 - x1)
So, plugging in the values, we get:
Slope = (15005 - 10237) / (2007 - 2005)
Slope = 4776 / 2
Slope = 2388
Thus, the estimated instantaneous rate of growth in 2006 is 2388. Ta-da!
d) Now, let's estimate the instantaneous rate of growth in 2007. We have the coordinates (2006, 12435) and (2008, 16684). Time to put on my math hat... er, hat that has math on it. Here we go!
Slope = (y2 - y1) / (x2 - x1)
Plugging in the values:
Slope = (16684 - 12435) / (2008 - 2006)
Slope = 4249 / 2
Slope = 2124.5
So, the estimated instantaneous rate of growth in 2007 is around 2124.5. Voilà!
e) Now let's compare these growth rates. In 2006, the estimated instantaneous rate of growth was 2388, while in 2007 it was approximately 2124.5. From this, we can conclude that the growth rate decreased slightly from 2006 to 2007. It's like when you eat a giant slice of pizza and then opt for a slightly smaller slice the next day - still growing, just at a slower pace.
To estimate the instantaneous rate of growth in 2006 (part c), we will need to measure the slope of the tangent line through the points (2005, 10237) and (2007, 15005).
To calculate the slope, we use the formula: slope = (change in y)/(change in x). In this case, the change in y is 15005 - 10237 = 4778, and the change in x is 2007 - 2005 = 2.
So the slope of the tangent line in 2006 is: slope = 4778/2 = 2389.
For part d, to estimate the instantaneous rate of growth in 2007, we need to measure the slope of the tangent line through the points (2006, 12435) and (2008, 16684).
Again, we calculate the slope using the formula slope = (change in y)/(change in x). The change in y is 16684 - 12435 = 4249, and the change in x is 2008 - 2006 = 2.
Therefore, the slope of the tangent line in 2007 is: slope = 4249/2 = 2124.5.
To compare the growth rates obtained in part c and part d (part e), we can conclude the following:
- In 2006 (part c), the instantaneous rate of growth is estimated to be approximately 2389.
- In 2007 (part d), the instantaneous rate of growth is estimated to be approximately 2124.5.
Comparing these two growth rates, we can conclude that the growth rate in 2006 (2389) is higher than the growth rate in 2007 (2124.5). This suggests that the company experienced a higher rate of growth in 2006 compared to 2007.
Slope of 2005 and 2007
(c) (15005-10237)/2 = 2384
(d) (16684-12435)/2 = 2124.5
(e) Looks like growth is slowing down