To determine the maximum number of gallons of water a family can use while keeping their monthly water bill under $37, we can set up the problem as follows:

1. The total water bill $B$ is given by the formula:

$B = 10 + 0.75 \times \text{(number of 1000 gallons used)}$

2. We want to find the maximum number of 1000-gallon units the family can use so that the bill is less than $37.

$10 + 0.75x < 37$

Where $x$ represents the number of 1000-gallon units used.

Now, solve for $x$:

1. Subtract 10 from both sides:

$0.75x < 27$

2. Divide both sides by 0.75:

$x < \frac{27}{0.75}$

3. Simplify the division:

$x < 36$

So, $x$ (the number of 1000-gallon units) must be less than 36.

Therefore, the family can use up to 35,999 gallons of water in a month to keep their bill under $37, since $x = 35.999$ (which is slightly less than 36 units, or 36,000 gallons).

Would you like more details, or do you have any questions?

Here are 5 related questions to explore further:

1. What would the water bill be if the family used exactly 36,000 gallons?
2. How would the maximum number of gallons change if the flat fee were increased to $15?
3. If the water rate increased to $1 per 1000 gallons, how many gallons could the family use while staying under $37?
4. What would be the impact on the water bill if the family reduced their usage by 10% each month?
5. How would the water bill be affected if there were an additional surcharge per gallon for exceeding a certain usage limit?

Tip: When dealing with cost-related problems, always isolate the variable representing the quantity (like the number of gallons) to easily determine the allowable usage within a budget.