To find the total cost to paint the floor of the circular swimming pool, follow these steps:

1. Calculate the radius of the pool: The diameter of the pool is 14 feet, so the radius $r$ is half of the diameter. $r = \frac{14}{2} = 7 \text{ feet}$

2. Calculate the area of the pool's floor: The area $A$ of a circle is given by the formula $A = \pi r^2$. $A = \pi \times 7^2 = \pi \times 49 = 153.938 \text{ square feet}$

3. Calculate the total cost to paint the pool's floor: The cost to paint 1 square foot is $5. $\text{Total cost} = 153.938 \times 5 = 769.69$

Since the options are given in whole numbers, we should round 769.69 to the nearest whole number, which is 770.

Therefore, the total cost to paint the pool's floor is:

$\boxed{770}$

Would you like more details or have any questions? Here are some more questions you might want to ask:

1. How do you derive the formula for the area of a circle?
2. How would the cost change if the diameter were different?
3. What if the cost per square foot to paint the pool changes?
4. How would the calculation change if the pool were an ellipse instead of a circle?
5. Can you explain the steps to calculate the area of irregular shapes?

Tip: Always ensure to double-check your calculations for accuracy, especially when rounding off values.