Problem 85 - Algebra and Trigonometry

Question

Driving cost It is estimated that the annual cost of driving a certain new car is given by the formula $$ C=0.35 m+2200 $$ where \(m\) represents the number of miles driven per year and \(C\) is the cost in dollars. Jane has purchased such a car and decides to budget between \(\$ 6400\) and \(\$ 7100\) for next year's driving costs. What is the corresponding range of miles that she can drive her new car?

Step-by-Step Solution

Answer

Jane can drive between 12000 and 14000 miles.

Step 1: Understand the Cost Formula

The formula given is \( C = 0.35m + 2200 \), where \( C \) represents the cost in dollars, and \( m \) is the number of miles driven in a year. Jane's budget constraints are between \( 6400 \) and \( 7100 \) dollars.

Step 2: Set Up Inequalities for the Budget

Using the budget constraints for \( C \), we set up two inequalities: 1. \( 6400 \leq 0.35m + 2200 \)2. \( 0.35m + 2200 \leq 7100 \).

Step 3: Solve the First Inequality

Solve \( 6400 \leq 0.35m + 2200 \):Subtract 2200 from both sides to get \( 6400 - 2200 \leq 0.35m \).This simplifies to \( 4200 \leq 0.35m \).Divide both sides by 0.35 to isolate \( m \):\( m \geq \frac{4200}{0.35} \),which simplifies to \( m \geq 12000 \).

Step 4: Solve the Second Inequality

Solve \( 0.35m + 2200 \leq 7100 \):Subtract 2200 from both sides to get \( 0.35m \leq 7100 - 2200 \).This simplifies to \( 0.35m \leq 4900 \).Divide both sides by 0.35 to solve for \( m \):\( m \leq \frac{4900}{0.35} \),which simplifies to \( m \leq 14000 \).

Step 5: Determine the Range of Miles

Combine the results from Steps 3 and 4 to find the range of miles:Jane can drive between 12000 miles and 14000 miles, or mathematically, \( 12000 \leq m \leq 14000 \).

Key Concepts

Inequalities in Algebra

Inequalities are a fundamental concept in algebra that help us determine the range of possible values that satisfy certain conditions.
They are similar to equations but instead of showing equality, they express that one side is either greater than or less than the other side. In the exercise, inequalities were used to define the budget constraints for Jane's driving costs.
Using the given cost formula, two inequalities were set up to represent the lower and upper limits of Jane's budget. Key points to remember when working with inequalities include:

• Always perform the same operations on both sides of the inequality.
• When multiplying or dividing both sides of an inequality by a negative number, the inequality sign is reversed.
• Rewriting inequalities into a simpler form can help in understanding the range of possible solutions.

By using inequalities, you can create a clear picture of all possible scenarios, like in Jane's case, where she needed to know how many miles she could drive without exceeding her budget.

Understanding Linear Equations

Linear equations form the backbone of solving real-world problems in algebra.
A linear equation is an equation between two variables that produces a straight line when plotted on a graph. This type of equation is characterized by its formula, typically in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
In Jane's situation, the equation \( C = 0.35m + 2200 \) represents a linear relationship between the cost \( C \) and the miles driven \( m \).Here are a few important aspects of linear equations:

• The slope \( m \) shows how much \( C \) changes for each additional mile driven.
• The y-intercept \( b = 2200 \) is the base cost that remains constant regardless of miles.
• Solving the equation involves isolating the variable, in this case \( m \), to find specific values.

Linear equations allow predictions and budgeting, as seen in Jane's budgeting decision based on her anticipated miles.

Budget Constraints in Algebra

Budget constraints are a practical application of algebra in financial planning and decision-making.
They represent the limits within which an individual must operate financially, such as the range Jane must adhere to in her driving costs.
Using inequalities constructed from a linear cost equation, one can assess the impact of varying one component, such as miles driven, on overall expenses. To effectively use budget constraints:

• Identify the key variables, such as costs and limits.
• Set up equations and inequalities to represent these relationships clearly.
• Solve the inequalities to find the acceptable range of choices.

In Jane's case, she defined her limits as $6400 and $7100, setting up inequalities that incorporated the car's cost equation to see what mileage falls within her budget. This approach ensures she doesn't surpass her financial boundaries.