Let the distance from Mr. Earl E. Bird's house to his workplace be denoted by \( d \) miles.
We start by analyzing the two scenarios provided:
-
When traveling at 40 mph:
- The time taken to travel the distance \( d \) is given by: \[ \text{Time} = \frac{d}{40} \text{ hours} \]
- Since Mr. Bird arrives 3 minutes late, he should have arrived at 8:00 A.M. + \( t \) to be on time, where \( t \) is the time he would take to arrive on time in hours.
- Converting 3 minutes to hours gives \( \frac{3}{60} = 0.05 \) hours.
- Thus, we have: \[ \frac{d}{40} = t + 0.05 \tag{1} \]
-
When traveling at 60 mph:
- The time taken at this speed is: \[ \text{Time} = \frac{d}{60} \text{ hours} \]
- In this scenario, he arrives 3 minutes early, meaning: \[ \frac{d}{60} = t - 0.05 \tag{2} \]
Now we have two equations (1) and (2): \[ \frac{d}{40} = t + 0.05 \] and \[ \frac{d}{60} = t - 0.05 \]
Next, we solve for \( t \) in both equations: From equation (1): \[ t = \frac{d}{40} - 0.05 \] From equation (2): \[ t = \frac{d}{60} + 0.05 \]
Setting the expressions for \( t \) equal to each other: \[ \frac{d}{40} - 0.05 = \frac{d}{60} + 0.05 \]
To eliminate the fractions, we can multiply the entire equation by 120 (the least common multiple of 40 and 60): \[ 120 \left( \frac{d}{40} \right) - 120 \cdot 0.05 = 120 \left( \frac{d}{60} \right) + 120 \cdot 0.05 \]
This simplifies to: \[ 3d - 6 = 2d + 6 \]
Now, let's isolate \( d \): \[ 3d - 2d = 6 + 6 \] \[ d = 12 \]
Now that we have the distance \( d \) (12 miles), we can calculate the time \( t \) it should take to reach on time: Using equation for \( t \): \[ t = \frac{d}{40} - 0.05 = \frac{12}{40} - 0.05 = 0.3 - 0.05 = 0.25 \text{ hours} \] This means Mr. Bird needs to arrive at his workplace in 0.25 hours.
To find the average speed \( s \) at which he should drive to arrive on time, we use the formula: \[ s = \frac{d}{t} = \frac{12 \text{ miles}}{0.25 \text{ hours}} = 48 \text{ miles per hour} \]
Thus, the average speed at which Mr. Bird should drive to arrive at his workplace precisely on time is: \[ \boxed{48} \text{ miles per hour} \]