In a certain factory, each day the expected number of accidents is related to the number of overtime hours by a linear equation. Suppose that on one day there were 1000 overtime hours logged and 8 accidents reported and on another day, there were 400 overtime hours logged and 5 accidents. What are the expected number of accidents when no overtime hours are logged?

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem, we need to establish the relationship between the number of overtime hours and the number of accidents using a linear equation. We will denote the number of overtime hours as \( X \) and the number of accidents as \( A \). The linear equation can be expressed as:

\[ A = mX + C \]

where \( m \) is the slope of the line and \( C \) is the y-intercept (the expected number of accidents when no overtime hours are logged).

### Step 1: Set up the equations based on the given data.

From the problem, we have two data points:
1. When \( X = 1000 \) hours, \( A = 8 \) accidents.
2. When \( X = 400 \) hours, \( A = 5 \) accidents.

We can set up the following equations based on these points:

1. \( 8 = 1000m + C \) (Equation 1)
2. \( 5 = 400m + C \) (Equation 2)

### Step 2: Subtract the two equations to eliminate \( C \).

To find \( m \), we can subtract Equation 2 from Equation 1:

\[ (8 - 5) = (1000m + C) - (400m + C) \]

This simplifies to:

\[ 3 = 600m \]

### Step 3: Solve for \( m \).

Now, we can solve for \( m \):

\[ m = \frac{3}{600} = \frac{1}{200} \]

### Step 4: Substitute \( m \) back into one of the original equations to find \( C \).

We can substitute \( m \) back into Equation 1 to find \( C \):

\[ 8 = 1000 \left(\frac{1}{200}\right) + C \]

This simplifies to:

\[ 8 = 5 + C \]

So,

\[ C = 8 - 5 = 3 \]

### Step 5: Write the final equation.

Now we have both \( m \) and \( C \):

\[ A = \frac{1}{200}X + 3 \]

### Step 6: Find the expected number of accidents when no overtime hours are logged.

When no overtime hours are logged, \( X = 0 \):

\[ A = \frac{1}{200}(0) + 3 = 3 \]

Thus, the expected number of accidents when no overtime hours are logged is **3**.

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