Problem 64 - Algebra and Trigonometry

Question

Fuel Consumption Suppose an automobile's fuel consumption is 28 \(\mathrm{mi} / \mathrm{gal}\) in city driving and 34 \(\mathrm{milgal}\) in highway driving. If \(x\) denotes the number of city miles and \(y\) denotes the number of highway miles, then the total miles this car can travel on a 15 -gallon tank of fuel must satisfy the inequality $$ \frac{1}{28} X+\frac{1}{34} y \leq 15 $$ Use this inequality to answer the following questions. (Assume that the car has a full tank of fuel.) (a) Can the car travel 165 city miles and 230 highway miles before running out of gas? (b) If the car has been driven 280 miles in the city, how many highway miles can it be driven before running out of fuel?

Step-by-Step Solution

Answer

(a) Yes, the car can travel 165 city miles and 230 highway miles. (b) The car can be driven 170 highway miles after 280 city miles.

Step 1: Understanding the Inequality

The given inequality \( \frac{1}{28}x + \frac{1}{34}y \leq 15 \) represents the fuel consumption of a car with a 15-gallon tank. Here, \(x\) stands for the city miles and \(y\) for highway miles. We need to check if specific mileages satisfy this inequality.

Step 2: Checking Part (a)

Substitute \(x = 165\) and \(y = 230\) into the inequality. Calculate \(\frac{165}{28} + \frac{230}{34}\) to see if it is less than or equal to 15. First, \(\frac{165}{28} \approx 5.8929\) and \(\frac{230}{34} \approx 6.7647\).Adding them gives approximately \(5.8929 + 6.7647 = 12.6576\) (rounded to four decimal places), which is less than 15. Therefore, the car can travel the given miles without running out of gas.

Step 3: Solving Part (b)

With \(x = 280\), find \(y\) when \(\frac{1}{28}x + \frac{1}{34}y = 15\). First, calculate \(\frac{280}{28} = 10\). The inequality becomes \(10 + \frac{y}{34} \leq 15\). To solve for \(y\), subtract 10 from both sides to get \(\frac{y}{34} \leq 5\).Multiply both sides by 34 to solve for \(y\), resulting in \(y \leq 170\). Therefore, the car can be driven up to 170 highway miles before running out of fuel.

Key Concepts

Fuel Consumption Calculation

When discussing fuel consumption calculation, it's crucial to understand how different driving conditions affect the total distance a vehicle can travel. Cars typically consume fuel at different rates depending on whether you're driving in the city or on the highway. This exercise blends these rates with basic algebra to determine if a set number of miles can be covered within a given fuel tank capacity.

• City driving often involves frequent stops and starts, resulting in higher fuel consumption per mile. Thus, a car's mile-per-gallon (mpg) will be lower in the city.
• Highway driving usually offers more consistent speeds, leading to better fuel efficiency and a higher mpg.

The exercise uses an inequality to represent these conditions mathematically. The inequality \( \frac{1}{28}x + \frac{1}{34}y \leq 15 \) allows for calculating fuel usage based on city (\(x\)) and highway (\(y\)) miles, given a 15-gallon tank. This form conveniently converts miles to gallons to see whether they fit within the tank's capacity.

Algebraic Problem Solving

Algebraic problem solving involves using mathematical expressions and inequalities to find unknown values. In this context, if you know how much fuel is available and how much each mile type consumes, you can solve for one variable given another. Let's break it down further.

• Substitute known values into the inequality to determine if a trip's mileage fits within the given fuel constraints. For example, check if specific city and highway mile combinations stay below the maximum allowable fuel usage.
• If the driven miles in one category are known, algebra allows solving for the other. This could involve simple operations such as addition, subtraction, multiplication, or division.

In the solution, we inspected whether 165 city miles and 230 highway miles fit within the tank capacity. Calculations showed that these values stayed within limits, affirming that the trip was feasible without exceeding fuel capacity. For solving the second challenge, applying basic algebra revealed the maximum highway miles possible after a set number of city miles.

City and Highway Mileage

Understanding city and highway mileage is essential for managing fuel efficiency and planning routes. Different driving environments offer variations in fuel consumption rates, and recognizing these helps optimize driving patterns.

• The city mileage rate; listed as 28 mpg, means the vehicle uses more fuel when driving within urban settings due to frequent idling and lower speeds.
• Conversely, at 34 mpg, highway mileage is more fuel-efficient, thanks to sustained speeds and less stop-and-go activity.

In practical terms, knowing that city driving uses more fuel per mile, one might choose a longer highway route over a shorter city path if it results in less overall fuel consumption. Given a full tank, using inequalities allows for projecting feasible travel distances for planned trips or detours. This exercise exemplifies how understanding mileage variations helps optimize fuel usage, keeping travel efficient and predictable.