Let's solve this step by step.

Step 1: Determine Apollo's initial speed.

Apollo traveled 48 leagues in 3 days. To find his speed (rate) in leagues per day:

$\text{Initial speed} = \frac{48 \text{ leagues}}{3 \text{ days}} = 16 \text{ leagues per day}$

Step 2: Determine his new speed after tripling his rate.

According to the problem, Apollo then tripled his rate. So, his new speed is:

$\text{New speed} = 3 \times 16 \text{ leagues per day} = 48 \text{ leagues per day}$

Step 3: Calculate the time needed to travel 336 leagues at the new speed.

Now, we need to find how long it will take him to travel 336 leagues at this new speed. Use the formula:

$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

Substitute the given values:

$\text{Time} = \frac{336 \text{ leagues}}{48 \text{ leagues per day}} = 7 \text{ days}$

So, it will take him 7 days to travel 336 leagues at the new speed.

Would you like more details or have any questions about this solution?

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Here are 5 related questions to consider:

1. If Apollo had only doubled his speed instead of tripling it, how long would it take him to travel 336 leagues?
2. What would have been Apollo's travel time if he maintained his original speed for the entire 336 leagues?
3. If Apollo had to travel 500 leagues at his new speed, how many days would it take?
4. How does the travel time change if Apollo's initial speed was 20 leagues per day?
5. If Apollo slowed down to half of his initial speed after the first 3 days, how would that affect his travel time for 336 leagues?

Tip: Always double-check the units (leagues, days, etc.) when solving problems involving speed and distance to ensure they are consistent throughout the calculations.