Problem 11 - Technical Mathematics with Calculus

Question

If your car now gets \(21.0 \mathrm{mi} / \mathrm{gal}\) of gas, and if you can go \(251 \mathrm{mi}\) on a tank of gasoline, how far could you drive with the same amount of gasoline with a car that gets \(35.0 \mathrm{mi} / \mathrm{gal} ?\)

Step-by-Step Solution

Answer

You could drive 587.5 miles with a car that gets 35.0 mi/gal on the same amount of gasoline.

Step 1: Determine the amount of gasoline in the tank

Since the car can go 251 miles on a full tank and the consumption rate is 21.0 miles per gallon, we can calculate the amount of gasoline in the tank by dividing the total distance by the consumption rate.

Step 2: Calculate the gallons of gasoline in the tank

Using the formula: \( \text{gallons of gasoline} = \frac{\text{total distance}}{\text{consumption rate}} \), we get \( \frac{251 \text{ mi}}{21.0 \text{ mi/gal}} \) to find the gallons in the tank.

Step 3: Calculate the distance with the new car

With the calculation from Step 2, we now have the amount of gasoline. To find out how far the new car can go with that amount of gasoline, multiply the gallons by the new consumption rate of 35.0 miles per gallon.

Step 4: Final Calculation

Multiply the gallons of gasoline by the new car's mileage to get the final distance: \( \text{gallons} \times 35.0 \text{ mi/gal} \)

Key Concepts

Mathematics Word Problems

Mathematics word problems are practical situations presented in a narrative form that require mathematical solutions. These problems involve using mathematical concepts and operations to solve real-life scenarios, like calculating distances, conversions, or budgets.

In the context of fuel efficiency, they often involve understanding the relationship between distance, fuel consumption, and efficiency. Tackling such problems necessitates careful reading to pick out the important numerical information and the relationships between them.

To improve understanding and solving mathematics word problems, practice identifying key terms and set up the calculation based on these terms. Visual representations, such as drawing diagrams or models, can also aid comprehension. Furthermore, doing estimation before calculating can help check if the final answer is reasonable.

Unit Conversion

Unit conversion is a crucial skill in mathematics and science that involves changing measurements from one unit to another. This is particularly important when dealing with international systems that may not use the same units for distance, volume, or efficiency.

In fuel efficiency problems, the ability to convert between units like miles to kilometers or gallons to liters is fundamental. Understanding unit conversion allows for more flexibility and accuracy when working with various types of data or inputs.

Common Unit Conversion Strategies

• Use conversion factors as a fraction to multiply and change units.
• Understand the relationship between units (e.g., 1 mile = 1.60934 kilometers).
• Memorize key conversion rates for frequent use.
• Double-check conversions with online tools or conversion charts.

When faced with different units in word problems, always convert to the necessary units before applying them to the formula.

Fuel Consumption Calculation

Fuel consumption calculation is the process of determining the amount of fuel required to cover a specific distance or the distance that can be traveled with a certain amount of fuel. This is expressed in terms like miles per gallon (mi/gal) or liters per 100 kilometers (L/100km), depending on the regional measurement preference.

Understanding how to calculate fuel consumption is vital for budgeting travel costs, comparing vehicle efficiency, or environmental impact assessments. For example, if a car travels 251 miles with 21.0 mi/gal, the amount of fuel used is found by dividing distance by fuel efficiency. Conversely, to find out how far a car can go with that amount of fuel at different efficiency rates, one should multiply the fuel quantity by the new efficiency rate.

Key Steps in Fuel Consumption Calculations

• Determine total distance and consumption rate.
• Calculate the amount of fuel used.
• Apply the new consumption rate to find the potential distance with the same fuel amount.

Engaging with fuel consumption problems helps in making more informed decisions regarding vehicle use and fuel budgeting. Estimating the potential travel distances under various fuel efficiencies is both a practical and environmentally relevant application of mathematical proficiency.