Question

Driving Cost It is estimated that the annual cost of driving a certain new car is given by the formula

$C=0.35m+2200$

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?

Short answer

The corresponding range of miles that she can drive her new car is between 12000 miles and 14000 miles (i.e) $12000\le m\le 14000$.

Step-by-step solution

Step 1. Apply the concept of Mathematical model, inequality.

A mathematical model is an equation that describes a real world object or process. Modeling is a process of finding such equations.

An inequality looks like an equation, except that in the place of the equal sign is one of the symbols, $<,>,\le ,\ge$.

Example of an inequality is $4x+7\le 19$.

An inequality is linear if each term is constant or a multiple of the variable. To solve a linear inequality, isolate the variable on one side of the inequality sign.

Step 2. Find the equations.

Let mbe the number miles driven per year and C be the cost in dollars

Hence, the following are obtained:

$\begin{array}{l}\text{6400=0.35m+2200------(1)}\\ \text{7100=0.35m+2200------(2)}\end{array}$

Step 3. Solve the equations.

From equation (1), the following is obtained:

$\begin{array}{c}6400=0.35m+2200\\ \Rightarrow 6400-2200=0.35m\\ \Rightarrow 4200=0.35m\\ \Rightarrow m=\frac{4200}{0.35}\\ \Rightarrow m=12000\end{array}$

Similarly, from equation (2), the following is obtained:

$\begin{array}{c}7100=0.35m+2200\\ \Rightarrow 7100-2200=0.35m\\ \Rightarrow 4900=0.35m\\ \Rightarrow m=\frac{4900}{0.35}\\ \Rightarrow m=14000\end{array}$

Therefore, the corresponding range of miles that she can drive her new car is between 12000 miles and 14000 miles.