Question

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.
rate of change: -10
this means the coal exports decreased 10 million tons per year between 2000 and 2002.

b. find the rate of change in coal exports between 2005 and 2006 and describe its meaning in the context of the situation.
rate of change: 10

Explanation:

Part a

Step 1: Identify data points

In 2000, exports (let's say \( y_1 \)) are 60 million short tons. In 2002, exports (\( y_2 \)) are 40 million short tons. Time difference \( \Delta x = 2002 - 2000 = 2 \) years.

Step 2: Calculate rate of change

Rate of change formula: \( \frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1} \)
Substitute values: \( \frac{40 - 60}{2002 - 2000}=\frac{-20}{2}=-10 \)
This means exports decreased by 10 million short tons per year (since rate is negative, indicating decrease) between 2000 and 2002.

Part b

Step 1: Identify data points

From the graph, in 2005 and 2006, assume exports are constant (since the line is flat). Let \( y_1 \) (2005) and \( y_2 \) (2006) be equal (say 45 million, but since change in \( y \) is 0). Time difference \( \Delta x = 2006 - 2005 = 1 \) year.

Step 2: Calculate rate of change

Rate of change: \( \frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1} \)
If \( y_2 = y_1 \), then \( \Delta y = 0 \), so rate of change is \( \frac{0}{1}=0 \)
This means coal exports remained constant (did not increase or decrease) between 2005 and 2006 (since rate is 0, no change per year).

Part a Answer:

Rate of change: \(-10\), meaning exports decreased 10 million short tons per year.

Part b Answer:

Rate of change: \(0\), meaning exports remained constant (no change) per year between 2005 and 2006.

Answer:

Step 1: Identify data points

From the graph, in 2005 and 2006, assume exports are constant (since the line is flat). Let \( y_1 \) (2005) and \( y_2 \) (2006) be equal (say 45 million, but since change in \( y \) is 0). Time difference \( \Delta x = 2006 - 2005 = 1 \) year.

Step 2: Calculate rate of change

Rate of change: \( \frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1} \)
If \( y_2 = y_1 \), then \( \Delta y = 0 \), so rate of change is \( \frac{0}{1}=0 \)
This means coal exports remained constant (did not increase or decrease) between 2005 and 2006 (since rate is 0, no change per year).

Part a Answer:

Rate of change: \(-10\), meaning exports decreased 10 million short tons per year.

Part b Answer:

Rate of change: \(0\), meaning exports remained constant (no change) per year between 2005 and 2006.