Problem 39 - Algebra Form and Function

Question

A car rental company charges \(\$ 37\) per day and \(\$ 0.25\) per mile. (a) Compute the cost of renting the car for one day, assuming the car is driven 100 miles. (b) Compute the cost of renting the car for three days, assuming the car is driven 400 miles. (c) Andy rented a car for five days, but he did not keep track of how many miles he drove. He gets a bill for \(\$ 385 .\) How many miles did he drive?

Step-by-Step Solution

Answer

Answer: The total cost for a one-day rental with 100 miles driven is $62, and the total cost for a three-day rental with 400 miles driven is $211. Andy drove 800 miles during his five-day rental.

Step 1: (a) Total cost for one day and 100 miles driven

First, we need to find the total cost for one-day rental, including the miles driven. The formula for the total cost is: Total Cost = (Daily Charge * Number of days) + (Per-mile Charge * Number of miles) For this situation, we have: Total Cost = (\(37 * 1) + (\)0.25 * 100) Now we compute the total cost: Total Cost = \(37 + \)25 = \(62\)

Step 2: (b) Total cost for three days and 400 miles driven

Similarly, for a three-day rental with 400 miles driven, we use the same formula for the total cost: Total Cost = (Daily Charge * Number of days) + (Per-mile Charge * Number of miles) For this situation, we have: Total Cost = (\(37 * 3) + (\)0.25 * 400) Now we compute the total cost: Total Cost = \(111 + \)100 = \(211\)

Step 3: (c) Determine miles driven with a given total cost and number of days rented

For this part, we are given the total cost and the number of days rented, and we need to find the number of miles driven. We can rearrange the total cost formula to solve for the number of miles: Number of miles = (Total Cost - (Daily Charge * Number of days)) / Per-mile Charge For this situation, we have: Number of miles = (\(385 - (\)37 * 5)) / $0.25 Now we compute the number of miles driven: Number of miles = (\(385 - \)185) / \(0.25 = \)200 / $0.25 = 800 Andy drove 800 miles.

Key Concepts

Algebraic Problem Solving

Algebraic problem solving involves using mathematical expressions and formulas to find unknown values. In everyday problems like the one involving car rental charges, we use algebra to calculate costs based on given rates. In this scenario, we are working with daily rental and mileage fees. By setting up an equation, we can insert known values to solve for unknown ones. For example:

• Daily charge is set at $37
• Per-mile charge is $0.25

To find the total cost for any number of days and miles, we use a formula that adds both daily and mileage costs. This approach allows us to apply algebraic methods flexibly to various situations, like calculating exact amounts spent or miles driven.

Cost Calculation

Cost calculation in this context involves combining fixed and variable charges. The fixed charge is the daily rental fee, while the variable charge depends on the number of miles driven. We apply the formula:\[\text{Total Cost} = (\text{Daily Charge} \times \text{Number of days}) + (\text{Per-mile Charge} \times \text{Number of miles})\]This formula is crucial because it gives us a straightforward way to determine how much renting a car will cost under different circumstances. For example:

• For 1 day and 100 miles, the cost is calculated as: \(37 \times 1 + 0.25 \times 100 = 62\)
• For 3 days and 400 miles, use: \(37 \times 3 + 0.25 \times 400 = 211\)

These calculations highlight the impact of both daily and mileage charges on the total cost, which can change depending on usage.

Variable Manipulation

In solving the problem of finding the number of miles driven, we use variable manipulation within a given equation. By rearranging our formula, we isolate the unknown variable, which in this case is the number of miles driven:\[\text{Number of miles} = \frac{(\text{Total Cost} - (\text{Daily Charge} \times \text{Number of days}))}{\text{Per-mile Charge}}\]Through this manipulation, we solve for the unknown by using known values:

• Total cost given as \(385
• 5 days of rental
• Daily charge of \)37
• Per-mile charge of $0.25

First, calculate the total daily cost and subtract it from the bill to find the mileage cost, then divide by the per-mile rate to find the miles. This technique is powerful for finding unknown values in any scenario where similar algebraic expressions apply.

Daily and Mileage Charges

Understanding daily and mileage charges helps in comprehending how rental costs accumulate. The daily charge is a flat fee that the company requires for each day the vehicle is rented. Each day adds a fixed cost, regardless of distance driven. Meanwhile, the mileage charge accumulates based on actual use, which aids those driving fewer miles in conserving costs.

• For example, Andy rented the car for 5 days, leading to a daily cost of \(37 \times 5 = 185\)
• This part of the cost is fixed regardless of how far Andy drives. But if he drives far, his mileage charge becomes significantly larger.

By understanding these charges, users can better plan their rental usage to balance between daily rents and mileage, potentially reducing overall rental costs if mileage is minimized.