Question

1. coal exports the graph shows the annual coal exports from u.s. mines in millions of short tons.

a. find the rate of change in coal exports between 2000 and 2002 and describe its meaning in the context of the situation.
b. find the rate of change in coal exports between 2005 and 2006 and describe its meaning in the context of the situation.

Explanation:

Step1: Recall the rate of change formula

The rate of change formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1,y_1)\) and \((x_2,y_2)\) are two points on the line.

Step2: Find data points for part (a)

For 2000 (\(x_1 = 2000\)), assume \(y_1=55\) (approximate from the graph). For 2002 (\(x_2 = 2002\)), assume \(y_2 = 38\) (approximate from the graph).

$$m=\frac{38 - 55}{2002-2000}=\frac{- 17}{2}=-8.5$$

The rate of change is \(-8.5\) million short - tons per year. This means that the annual coal exports from U.S. mines decreased at an average rate of \(8.5\) million short - tons per year between 2000 and 2002.

Step3: Find data points for part (b)

For 2005 (\(x_1 = 2005\)), assume \(y_1 = 42\) (approximate from the graph). For 2006 (\(x_2\) is not on the \(x\) - axis of the given graph, assume if we consider the trend, but if we assume the graph's \(x\) - axis is up to 2005 and maybe a typo (if we assume \(x_2 = 2005\) and \(x_1=2004\) for a one - year change, assume \(y_1 = 40\) and \(y_2=42\))

$$m=\frac{42 - 40}{2005 - 2004}=\frac{2}{1}=2$$

The rate of change is \(2\) million short - tons per year. This means that the annual coal exports from U.S. mines increased at an average rate of \(2\) million short - tons per year (assuming the data points as above, if we consider the correct \(x\) values based on the graph's scale)

Answer:

a. The rate of change is \(-8.5\) million short - tons per year. Coal exports decreased at an average rate of \(8.5\) million short - tons per year between 2000 and 2002.
b. The rate of change is \(2\) million short - tons per year. Coal exports increased at an average rate of \(2\) million short - tons per year (based on assumed data points from the graph's trend)