Let's define the speeds of the two cars and use the information given to form equations.

1. Let the speed of the slower car be $x$ mph.
2. Then, the speed of the faster car is $x + 8$ mph.

Since they are moving toward each other, the total distance they cover when they meet is the sum of the distances each car travels in 2.5 hours. This total distance is given as 310 miles.

Step 1: Set Up the Equation

The distance each car travels is given by its speed multiplied by the time. Since the cars meet in 2.5 hours, the combined distance equation is:

$x \cdot 2.5 + (x + 8) \cdot 2.5 = 310$

Step 2: Simplify the Equation

Distribute $2.5$ to both terms:

$2.5x + 2.5(x + 8) = 310$

Combine like terms:

$2.5x + 2.5x + 20 = 310$

$5x + 20 = 310$

Step 3: Solve for $x$

Subtract 20 from both sides:

$5x = 290$

Then, divide by 5:

$x = 58$

Step 4: Find the Speed of Each Car

• The slower car’s speed is $x = 58$ mph.
• The faster car’s speed is $x + 8 = 58 + 8 = 66$ mph.

Answer

• The slower car's speed is 58 mph.
• The faster car's speed is 66 mph.

Would you like further explanation or details on any part?

Here are some additional related questions to explore:

1. What if the cars had started 500 miles apart? How would that change their speeds?
2. How long would it take for the cars to meet if the faster car was only 5 mph faster?
3. What if one car started 30 minutes after the other? How would you set up the problem?
4. If one car was driving 50 mph, what would be the speed of the other car?
5. How would the solution change if they were traveling away from each other instead?

Tip: For problems involving relative speeds, remember that when two objects move toward each other, you add their speeds, while if they move away, you subtract.