Problem 11 - Algebra 1

Question

Suppose that bike rentals cost \(\$ 4\) plus \(\$ 1.50\) per hour. Use the equation to find the cost of renting a bike for 12 hours.

Step-by-Step Solution

Answer

The cost of renting a bike for 12 hours is \$22.

Step 1: Setting Up The Equation

To find the cost for renting a bike for 12 hours, first set up the equation based on the given information. Let \( C \) be the total cost of renting a bike for \( t \) hours. Then the equation can be written as \( C = 4 + 1.5t \).

Step 2: Substituting The Value

Substitute the value \( t = 12 \) into the equation which gives \( C = 4 + 1.5*12 \).

Step 3: Carrying Out The Calculation

Carry out the calculation on the right-hand side to get the total cost. \( C = 4 + 18 = 22 \)

Key Concepts

Cost Calculation

When calculating costs based on an hourly rate plus a fixed fee, you integrate both parts into an equation. In the case of bike rentals, there's a fixed base cost of \(4 plus a variable cost of \)1.50 for each additional hour. This structure is common in many service industries where both base fees and usage fees are present. To determine the cost for a specific time length, like 12 hours, you insert the hours into your formula. The formula, in this case, is written as \( C = 4 + 1.5t \), where \( C \) represents the cost and \( t \) represents the total hours rented.

When you calculate this, remember to:

• Include the base cost of $4 in every calculation.
• Multiply the hourly rate by the total hours to find the variable cost.
• Add both parts together to get the final cost.

This method ensures all aspects of the rate are accounted for, leading to an accurate total cost.

Substitution Method

The substitution method is a strategy used in algebra to solve equations by substituting one value in place of a variable. In the exercise, we use this method to determine the cost of renting a bike for a specific number of hours.

Here's how it works:

• First, identify your equation: \( C = 4 + 1.5t \).
• Next, substitute the known value of \( t \), which is 12, into the equation. This transforms it into \( C = 4 + 1.5 \times 12 \).
• With the value of \( t \) replaced, you can now calculate \( C \) to find the total cost.

This method is handy as it simplifies the process by eliminating the variable and directly computing with known values. It brings clarity and precision, especially in problems involving clear numerical setups.

Linear Equations

Linear equations are fundamental in algebra and often appear in cost calculations. A linear equation forms a straight line when plotted on a graph and has a consistent rate of change. The general form of a linear equation is \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.

The bike rental cost equation \( C = 4 + 1.5t \) is a linear equation, where:

• \( C \) is analogous to y, the total cost.
• \( 1.5 \) is the slope, representing the hourly rate.
• \( 4 \) is the y-intercept, indicating the base cost when no hours are used.

Linear equations help maintain a predictable and manageable relationship between variables, making it easy to calculate and predict costs for different scenarios. Understanding these concepts allows one to anticipate how changes in one variable (like hourly usage) can affect the total cost.