Question

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 scatterplot shows her results. based on her line of best fit, which of the following statements is true?
a. the initial number of cracks was 2.
b. the initial number of cracks was 10.
c. for each bolt used, the number of cracks decreased by an average of ½.
d. for each bolt used, the number of cracks decreased by an average of 3.

Explanation:

Step1: Find line slope (rate change)

Use points (2,8) and (8,5):
$$\text{Slope} = \frac{5-8}{8-2} = \frac{-3}{6} = -\frac{1}{2}$$

Step2: Verify y-intercept (initial cracks)

Use slope and (2,8) in $y=mx+b$:
$$8 = -\frac{1}{2}(2) + b \implies 8 = -1 + b \implies b=9$$

Step3: Evaluate each option

• A: Initial cracks = 9 ≠ 2 (false)
• B: Initial cracks = 9 ≠ 10 (false)
• C: Slope $-\frac{1}{2}$ means cracks decrease by $\frac{1}{2}$ per bolt (true)
• D: Slope is $-\frac{1}{2}$ ≠ -3 (false)

Answer:

C. For each bolt used, the number of cracks decreased by an average of ½.