Problem 17 - Algebra 1: Concepts and Skills

Question

In Exercises \(12-17\), use the following information. Renting a canoe costs 10 dollars plus 28 dollars per day. The linear model for this situation relates the total cost of renting a canoe, \(y,\) with the number of days rented, \(x\). If you had 66 dollars to spend, for how many days could you rent a

Step-by-Step Solution

Answer

With 66 dollars, you could rent a canoe for 2 days.

Step 1: Understand the equation

There is a given linear model for the total cost of renting a canoe: \( y = 10 + 28x \), where \( y \) is the total cost and \( x \) is the number of days rented.

Step 2: Substitute the known variables

The problem states that you have 66 dollars to spend. Substitute this value for \( y \) in the equation: \( 66 = 10 + 28x \).

Step 3: Solve for \( x \)

To solve for \( x \), first subtract 10 from both sides of the equation: \( 56 = 28x \). Then, divide both sides by 28 to solve for \( x \): \( x = 2 \).

Key Concepts

Algebra

Algebra is a branch of mathematics that helps us in solving equations. It involves using symbols and letters to represent numbers and variables.

• In the exercise, the linear equation given is in the form of a simple algebraic equation where the total cost (\( y \)) depends on the number of days rented (\( x \)).
• We see a fixed cost (10 dollars) which you pay regardless of the number of days you rent.
• Every additional day comes with an additional cost of 28 dollars.

Understanding the structure of these equations allows us to see how each part contributes to the total cost. By setting up the equation correctly, we can then manipulate the terms to derive the unknown variable, in this case, the number of days, using simple steps.

Problem Solving

Problem solving involves identifying the problem, organizing the given information, and systematically solving the equation.

• The problem in this exercise is determining how many days you can rent a canoe with a limited budget of 66 dollars.
• To solve like this, it's important to break it into clear steps.
• First, recognize that 66 dollars is the total amount available, representing the total cost in the equation.

Then substitute this value into the equation: First, identify the cost equation as given: \( y = 10 + 28x \).Next, substitute 66 for \( y \) since that's the budget we are working with: \(66 = 10 + 28x\). Each of these steps, when done carefully, helps to arrive at the solution logically and efficiently.

Cost Calculation

Cost calculation helps determine total expenses based on usage, such as how we calculate costs of renting.

• In this scenario, the cost is determined by a base rental fee plus a daily rental rate.
• The linear equation that describes this relationship is \( y = 10 + 28x \), where 10 dollars is a fixed fee, and 28 dollars is a daily fee.

This means for each day you rent the canoe, you'll add 28 dollars to the starting cost of 10 dollars.The calculation process involves solving the equation for \( x \). First, subtract the base fee (10 dollars) from the total budget (66 dollars) to get the amount available for daily fees: \( 66 - 10 = 56 \).Then, divide the remaining amount by the daily cost to find the number of days: \( 56 / 28 = 2 \).Thus, you can rent the canoe for 2 days within a budget of 66 dollars.