Question

question 1 of 13 for a series of concrete structures, a civil engineer records the number of cracks that develop in each one and the number of fastening bolts used in that structure. the scatter - plot shows her results. based on the line of best fit, which statement is true? a. for each bolt used, the number of cracks decreased by an average of 1/2. b. the initial number of cracks was 10. c. the initial number of cracks was 8. d. for each bolt used, the number of cracks decreased by an average of 3.

Explanation:

Step1: Recall slope - intercept form

The equation of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. In the context of the scatter - plot, $x$ is the number of bolts, $y$ is the number of cracks, $m$ represents the change in the number of cracks per change in the number of bolts, and $b$ is the initial number of cracks when no bolts are used.

Step2: Calculate the slope

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. Using the points $(2,8)$ and $(8,5)$:
$m=\frac{5 - 8}{8 - 2}=\frac{-3}{6}=-\frac{1}{2}$
This means that for each increase of 1 in the number of bolts (increase in $x$), the number of cracks ( $y$) decreases by an average of $\frac{1}{2}$.

Step3: Analyze the y - intercept

To find the y - intercept, we can use the point - slope form $y - y_1=m(x - x_1)$ and then set $x = 0$. But we can also observe from the graph that the line does not intersect the y - axis at 8 or 10.

Answer:

A. For each bolt used, the number of cracks decreased by an average of $\frac{1}{2}$.