# Graphs

## 9/27 Graphs

Read the following SAT test question and then select the correct answer.

When a question includes a graph, it is especially important to read both the text under the graph and the labels on the graph. Identify the bottom line and assess your options for reaching it.  Ask yourself, "What could I do?" and then "What should I do?"  Once you have selected an efficient method to solve the problem, attack the problem!  Loop back to make sure that your answer addresses the bottom line.

The histogram above shows the distribution of 31 black cherry trees, by height. For example, the leftmost bar represents the black cherry trees that are at least 60 feet, but not more than 65 feet, in height. Based on the histogram, which of the following can be the average (arithmetic mean) height of the 31 black cherry trees?

Your bottom line is the average height of 31 trees, not the exact average, but what it could be.  This histogram does not tell you the exact height of any of the trees, so how can you find their average heights?  Look at that first  bar.  There are three trees that must be between 60 and 65 feet in height.  If you assume that all of those trees are as short as possible (60 feet), you will find the lowest value that their  average could possibly be.  Find the lowest height that all of the trees could possibly be and then average those heights together.

$\frac{3(60)+3(65)+8(70) + 10(7.5)+5(80)+2(85)}{31}\approx 72.74$

The lowest possible average for the heights of these trees is 72.74.  Any answer lower than this will be wrong.  Now, you could go back into your equation and plug in the highest possible value for each tree and average them again, but that will take a lot of time to retype into your calculator.  Instead, you should think logically about the height of the trees.  If you use the highest height that any tree can be, you are adding 5 to every single tree on the chart.  That means that your final average will be 5 feet higher than your current average.

$72.74 + 5 = 77.74$

You now have the highest and lowest possible averages of the heights of the trees.  Since your bottom line asks which of the following answers could be the average, you must eliminate any answers that are not between 72.74 and 77.74.

(A) 70 feet

(B) 72 feet
(C) 74 feet
(D) 78 feet
(E) 80 feet