Let's solve the problem:

Step 1: Define Variables

• Let the speed of the slower automobile be $x$ mph.
• Then, the speed of the faster automobile will be $x + 10$ mph.

Step 2: Write an Equation for Total Distance

After 2 hours, the total distance traveled by both automobiles combined is 160 miles.

• The slower car travels $2x$ miles in 2 hours.
• The faster car travels $2(x + 10)$ miles in 2 hours.

Since they are moving in opposite directions, the sum of the distances they cover equals 160 miles:

$2x + 2(x + 10) = 160$

Step 3: Simplify and Solve for $x$

$2x + 2x + 20 = 160$ $4x + 20 = 160$ $4x = 160 - 20$ $4x = 140$ $x = 35$

So, the slower car is traveling at 35 mph.

Step 4: Find the Speed of the Faster Automobile

$x + 10 = 35 + 10 = 45 \, \text{mph}$

Step 5: Verify the Answer

• In 2 hours, the slower car covers $2 \times 35 = 70$ miles.
• The faster car covers $2 \times 45 = 90$ miles.
• Total distance: $70 + 90 = 160$ miles, which matches the problem statement.

Final Answer

The two cars are traveling at:

• 35 mph and 45 mph.

Thus, the correct option is (d) 35 mph; 45 mph.

Do you want further clarification or have additional questions?

Here are 5 related questions:

1. What if the two cars traveled for 3 hours instead? How far apart would they be?
2. How would the equation change if one car was 15 mph faster than the other?
3. How do you convert speeds between mph and kph?
4. What is the relative speed of two objects traveling in opposite directions?
5. How would you solve this problem if only one car's speed was given?

Tip: Problems involving relative speed can be simplified by combining their speeds when objects move toward or away from each other.