Question

In a factory, the expected number of accidents per day is linearly related to the overtime hours \(x\). On a day with \(x = 1000\) overtime hours there were 8 accidents; on a day with \(x = 400\) hours there were 5 accidents. What is the expected number of accidents when no overtime is logged (\(x = 0\))?

• A. 2
• B. 3
• C. 4
• D. 5

Hint:

Model accidents as a straight line \(A = ax + c\) in overtime hours \(x\); \(c\) is exactly the value at \(x=0\) the question wants.

Answer 1

The Correct Option is B

Step 1: Find the rate of accidents per overtime hour.
Between the two given points, overtime hours change from 400 to 1000, a change of $600$ hours, while accidents change from 5 to 8, a change of $3$ accidents. Since the relationship is a straight line, this ratio is the same everywhere on the line:
$$ \text{rate} = \frac{8-5}{1000-400} = \frac{3}{600} = \frac{1}{200} \text{ accidents per overtime hour} $$

Step 2: Walk back from a known point to $x=0$.
Take the point $(400, 5)$. To reach $x = 0$, go back $400$ hours. Going back along a straight line means subtracting $400$ times the rate from the accident count:
$$ A(0) = 5 - 400 \times \frac{1}{200} = 5 - 2 = 3 $$

Step 3: Cross check with the other point.
Starting instead from $(1000, 8)$ and walking back $1000$ hours at the same rate:
$$ A(0) = 8 - 1000 \times \frac{1}{200} = 8 - 5 = 3 $$
Both starting points give the same value, confirming the line is consistent.

Step 4: Conclusion.
With zero overtime hours logged, the factory would still expect 3 accidents a day on average, which is option (B).
$$ \boxed{3} $$