Locate 3 on the x axis number line. Draw a vertical line through 3 and this vertical line will cross the parabola at some point P. Mark this point P on the parabola. Then draw a horizontal line from P to the y axis. The horizontal line will land on y = 10. In short, this all shows us that (3,10) is a point on this parabola.
Repeat those steps above, but now for x = 7. You'll see that (7,90) is another point on this parabola.
We need to find the slope of the line through the two points (3,10) and (7,90). The average rate of change from x = 3 to x = 7 is the same as the slope of the line through those two points.
To find the slope, we use the slope formula m = (y2 - y1)/(x2 - x1) where (x1,y1) and (x2,y2) are the two points, and m is the slope
In this case, (x1,y1) = (3,10) and (x2,y2) = (7,90) further breaking down to x1=3 y1=10 x2=7 y2=90 So we'll plug those four pieces of info into the equation and simplifying to get... m = (y2 - y1)/(x2 - x1) m = (90 - 10)/(7 - 3) m = 80/4 m = 20
The slope of the line is 20, so therefore, the average rate of change is 20.
The height of the tree is the sum of the part below the line parallel to the horizontal and the part above the line parallel to the horizontal.
Height of the part below the line parallel to the horizontal = 18 sin16° = 4.96 meters Horizontal distance of the tip of the of the shadow from the tree = 18 cos16° = 17.30 meters Height of the part above the line parallel to the horizontal = 17.3 tan68° = 42.83 meters
