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Let's solve the problem step by step using the information and setup provided in the image.

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Problem Summary:

- Samy leaves town driving at 40 mph.
- 2 hours later, Carol leaves from the same place, traveling the same road.
- Carol catches up to Samy after 5 hours and 20 minutes of her own travel time.
- We are to find Carol’s speed, denoted as $ x $ mph.

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Step 1: Understand the times

We need to be careful about the time each person has been driving when Carol catches up.

- Carol drives for:
$ 5 \text{ hours } 20 \text{ minutes} = 5 + \frac{20}{60} = 5 + \frac{1}{3} = \frac{16}{3} $ hours

- Samy has been driving for:
She left 2 hours earlier, so she has driven:
$$
2 + \frac{16}{3} = \frac{6}{3} + \frac{16}{3} = \frac{22}{3} \text{ hours}
$$

But wait — look at the diagram. It says:

> Samy has driven $ 5\frac{1}{3} $ hours, and Carol has driven $ 3\frac{1}{3} $ hours.

Wait! That contradicts what we just calculated.

Let’s double-check.

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Re-examining the Diagram:

The diagram says:

- Carol drives for $ 3\frac{1}{3} $ hours (i.e., $ 3 + \frac{1}{3} = \frac{10}{3} $ hours)
- Samy drives for $ 5\frac{1}{3} $ hours (i.e., $ \frac{16}{3} $ hours)

But if Carol left 2 hours later, then:

$$
\text{Samy's time} = \text{Carol's time} + 2
$$

Let’s test:

If Carol drives $ 3\frac{1}{3} = \frac{10}{3} \approx 3.33 $ hours, then Samy has driven:

$$
\frac{10}{3} + 2 = \frac{10}{3} + \frac{6}{3} = \frac{16}{3} = 5\frac{1}{3} \text{ hours}
$$

✔ That matches the diagram.

So the problem statement says: *"She catches Samy after 5 hours and 20 minutes"*

But 5 hours and 20 minutes = 5 + 1/3 = 16/3 ≈ 5.33 hours, which is Samy’s total time, not Carol’s.

Ah! So the phrase "after 5 hours and 20 minutes" refers to how long after Carol started, or how long Samy has been driving?

Let’s read carefully:

> "She catches Samy after 5 hours and 20 minutes."

This is ambiguous, but given that the diagram shows Carol driving $ 3\frac{1}{3} $ hours and Samy $ 5\frac{1}{3} $, and since Carol left 2 hours later, it makes sense that Carol drives for $ 3\frac{1}{3} $ hours, and Samy drives for $ 5\frac{1}{3} $ hours.

So the "after 5 hours and 20 minutes" must refer to the time since Carol left? But 5h 20m = $ 5\frac{1}{3} $, not $ 3\frac{1}{3} $.

There’s a contradiction.

Wait — let’s re-read the original text:

> "2 hours later Carol leaves... She catches Samy after 5 hours and 20 minutes."

So "after 5 hours and 20 minutes" — after what? After Carol leaves?

Yes — likely: Carol catches Samy 5 hours and 20 minutes after she (Carol) starts.

But that would mean Carol drives $ 5\frac{1}{3} $ hours, and Samy drives $ 5\frac{1}{3} + 2 = 7\frac{1}{3} $ hours.

But the diagram says Carol drives $ 3\frac{1}{3} $ hours and Samy $ 5\frac{1}{3} $.

So either the diagram or the wording is inconsistent.

Let’s compare both interpretations.

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Option 1: Assume the diagram is correct

Diagram says:

- Carol drives $ 3\frac{1}{3} $ hours → $ \frac{10}{3} $ hrs
- Samy drives $ 5\frac{1}{3} $ hours → $ \frac{16}{3} $ hrs
- Difference: $ \frac{16}{3} - \frac{10}{3} = \frac{6}{3} = 2 $ hours → ✔ matches “2 hours later”

So Carol drives for $ 3\frac{1}{3} $ hours, and Samy drives for $ 5\frac{1}{3} $ hours.

But the text says: "She catches Samy after 5 hours and 20 minutes"

5h 20m = $ 5 + \frac{1}{3} = \frac{16}{3} $ hours → this is Samy’s total time, not Carol’s.

So the phrase "after 5 hours and 20 minutes" must mean after Samy started, not after Carol.

But that’s awkward. Usually, “she catches Samy after X time” means from when she started.

Alternatively, perhaps the diagram is wrong, or the text is misphrased.

But the equation in the diagram is:

$$
(40)\left(5\frac{1}{3}\right) = (x)\left(3\frac{1}{3}\right)
$$

That suggests they are setting distances equal.

Let’s go with the diagram’s logic, since it includes the equation.

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Assume:

- Samy drives for $ 5\frac{1}{3} $ hours at 40 mph
- Carol drives for $ 3\frac{1}{3} $ hours at $ x $ mph
- They travel the same distance when Carol catches up.

So:

$$
\text{Distance}_\text{Samy} = \text{Distance}_\text{Carol}
$$

$$
(40) \times \left(5\frac{1}{3}\right) = x \times \left(3\frac{1}{3}\right)
$$

Convert mixed numbers to improper fractions:

- $ 5\frac{1}{3} = \frac{16}{3} $
- $ 3\frac{1}{3} = \frac{10}{3} $

So:

$$
40 \times \frac{16}{3} = x \times \frac{10}{3}
$$

Multiply both sides by 3 to eliminate denominators:

$$
40 \times 16 = x \times 10
$$

$$
640 = 10x
$$

$$
x = \frac{640}{10} = 64
$$

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✔ Answer: Carol was driving at 64 mph.

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Check:

- Samy drives $ 5\frac{1}{3} = \frac{16}{3} $ hours at 40 mph:
$$
d = 40 \times \frac{16}{3} = \frac{640}{3} \approx 213.33 \text{ miles}
$$

- Carol drives $ 3\frac{1}{3} = \frac{10}{3} $ hours at 64 mph:
$$
d = 64 \times \frac{10}{3} = \frac{640}{3} \approx 213.33 \text{ miles}
$$

✔ Same distance.

Also, Carol left 2 hours later, so she drives $ \frac{10}{3} \approx 3.33 $ hours, while Samy drives $ \frac{16}{3} \approx 5.33 $ hours → difference is 2 hours → ✔

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Conclusion:

Even though the wording "after 5 hours and 20 minutes" might seem confusing, based on the diagram and consistent math, Carol drove for $ 3\frac{1}{3} $ hours, and the "5 hours and 20 minutes" refers to Samy’s total driving time.

Thus, Carol’s speed is $ \boxed{64} $ mph.

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✔ Final Answer: Carol was driving at 64 mph.