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Chapter 5: Problem 14

APPLICATION Automobile companies advertise two rates for fuel mileage. City mileage is the rate of fuel consumption for driving in stop-and-go tra Cynthia's new car gets \(17 \mathrm{mi} /\) gal in the city and \(25 \mathrm{mi} / \mathrm{gal}\) on the highway. She drove 220 miles on 11 gallons of gas. a. Define variables and write a system of equations for the gallons burned at each mileage rate. (a) b. Solve this system and explain the meaning of the solution. (a) c. Find the number of city miles and highway miles Cynthia drove. d. Check your answers. (i)

Short Answer

Expert verified
Cynthia drove 116.875 city miles and 103.125 highway miles.

Step by step solution

01

Define Variables

Let \(c\) be the gallons burned in the city and \(h\) be the gallons burned on the highway. These represent the fuel consumption at each mileage rate.
02

Form Equations from Mileage Rates

The total miles driven was 220, thus we have the equation: \[ 17c + 25h = 220 \] because the fuel efficiency is 17 mi/gal in the city and 25 mi/gal on the highway.
03

Form Equation from Total Gallons

The total gallons consumed is 11, so we have another equation: \[ c + h = 11 \] which represents the total fuel used.
04

Solve the System of Equations

We have the system of equations: \[ \begin{align*} 17c + 25h &= 220 \ c + h &= 11 \end{align*} \] Solve for one of the variables in the second equation, \(c = 11 - h\), and substitute into the first: \[17(11 - h) + 25h = 220\]Expand and simplify: \[187 - 17h + 25h = 220 \] Combine like terms: \[ 8h = 33 \] So, \(h = \frac{33}{8} = 4.125\).
05

Find Amount of Gallons Burned in the City

Substitute \(h = 4.125\) back into the equation \(c + h = 11\),so \(c = 11 - 4.125 = 6.875\).
06

Calculate City and Highway Miles

City miles: \(17 \times 6.875 = 116.875\) milesHighway miles: \(25 \times 4.125 = 103.125\) miles.
07

Verification (Check the Total Distance)

Verify by adding the calculated city and highway miles: \(116.875 + 103.125 = 220\) miles, confirming the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fuel Efficiency
Fuel efficiency measures how far a vehicle can travel on a certain amount of fuel. For Cynthia鈥檚 car, the city mileage is 17 miles per gallon (mi/gal), and the highway mileage is 25 miles per gallon. This means that for every gallon of fuel, her car can travel 17 miles in the city and 25 miles on the highway.
Understanding fuel efficiency is important for calculating fuel costs and planning journeys. In this exercise, it helps us determine how many miles Cynthia drove in the city versus on the highway. Knowing these rates and the total gallons Cynthia used provides a basis for setting up our system of equations.
Algebraic Problem Solving
Algebraic problem solving involves using mathematical equations to find unknown values. In this scenario, we鈥檙e tasked with figuring out how much fuel was used in the city and on the highway.
To tackle this, we define variables: let \( c \) be the gallons burned in the city and \( h \) the gallons burned on the highway. By forming equations based on the given fuel efficiencies and total fuel usage, we create a system of equations that when solved, reveals the quantities we're seeking. The key is to use algebraic manipulation to isolate variables, making it easier to find precise answers.
Linear Equations
Linear equations are equations of the first degree, meaning they involve only sums and differences of variables raised to the power of one. In this problem, two linear equations are created:
  • \( 17c + 25h = 220 \): represents the total miles driven using city and highway mileages.
  • \( c + h = 11 \): indicates the total gallons of fuel burned.
We solve these equations simultaneously to find the values of \( c \) and \( h \). This involves substitution or elimination methods, where one equation is manipulated to express one variable in terms of the other, allowing us to uncover exact values when plugged back into the system.
Distance Calculation
Distance calculation stems from using the variables and efficiency rates to find the actual miles covered. Once we know \( c \) and \( h \) 鈥� the gallons burned in the city and on the highway 鈥� we calculate the distances. For Cynthia:
  • City miles: \( 17 \times 6.875 = 116.875 \) miles
  • Highway miles: \( 25 \times 4.125 = 103.125 \) miles
Adding these results confirms Cynthia drove exactly 220 miles, the total distance given in the problem. This calculation step is crucial as it verifies the solution and ensures every piece of information aligns perfectly with the initial data provided.

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