Solution for Part (a)

Expressions to estimate the amount they will pay for gasoline and the number of stops for a trip of $m$ miles

1. Fuel Efficiency: The car gets between 25 and 29 miles per gallon.
   Average fuel efficiency can be estimated as: $\text{Average MPG} = \frac{25 + 29}{2} = 27 \, \text{miles per gallon}.$

2. Fuel Consumption: For a trip of $m$ miles, the number of gallons of gas required is: $\text{Gallons required} = \frac{m}{\text{MPG}}.$

   Using the extremes of MPG (25 and 29), the range of gallons required is: $\text{Gallons (min)} = \frac{m}{29}, \quad \text{Gallons (max)} = \frac{m}{25}.$

3. Number of Stops: The car's fuel tank holds 17 gallons, and they refill the tank when $\frac{1}{4}$ full. Thus, they use $\frac{3}{4}$ of the tank capacity before stopping: $\text{Gallons per stop} = 17 \times \frac{3}{4} = 12.75 \, \text{gallons}.$

   The number of stops can be calculated as: $\text{Stops} = \frac{\text{Gallons required}}{12.75}.$

4. Cost of Gasoline: Gas costs between $3.47 and $3.69 per gallon. Using the range of gallons required, the total cost is: $\text{Cost (min)} = \frac{m}{29} \times 3.47, \quad \text{Cost (max)} = \frac{m}{25} \times 3.69.$

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Solution for Part (b)

Given Information:

• Distance from Denver to Chicago: $1004 \, \text{miles}$.
• Round trip distance: $\text{Total distance} = 1004 \times 2 = 2008 \, \text{miles}.$

1. Gallons Required: Using the MPG range: $\text{Gallons (min)} = \frac{2008}{29}, \quad \text{Gallons (max)} = \frac{2008}{25}.$

   Calculations: $\text{Gallons (min)} \approx 69.24, \quad \text{Gallons (max)} \approx 80.32.$

2. Number of Stops: Using $12.75 \, \text{gallons per stop}$: $\text{Stops (min)} = \frac{69.24}{12.75} \approx 5.43 \Rightarrow 6 \, \text{stops (rounding up)}.$ $\text{Stops (max)} = \frac{80.32}{12.75} \approx 6.3 \Rightarrow 7 \, \text{stops (rounding up)}.$

3. Cost of Gasoline: Using gas prices of $3.47 to $3.69 per gallon: $\text{Cost (min)} = 69.24 \times 3.47 \approx 240.28, \quad \text{Cost (max)} = 80.32 \times 3.69 \approx 296.38.$

   Therefore: $\text{Total cost range: } \$240.28 \, \text{to} \, \$296.38.$

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Final Answers

(a):

• Number of stops (expression): $\frac{\frac{m}{\text{MPG}}}{12.75}$.
• Cost (expression): $\frac{m}{\text{MPG}} \times \text{Gas Price Range}$.

(b):

• Total cost: $240.28 to $296.38.
• Number of stops: 6 to 7.

Would you like further clarification or calculations? Here are related questions for deeper understanding:

1. How is fuel efficiency factored into budgeting trips?
2. What is the role of tank capacity in estimating stops?
3. How can variations in gas prices affect trip planning?
4. Why is rounding up the number of stops necessary?
5. How does a range of MPG values provide flexibility in calculations?

Tip: Always calculate using both minimum and maximum estimates to account for uncertainties!