Problem 26 - Elementary Algebra

Question

The relationship of the time a tour guide works, \(x\), and the cost to hire the tour guide, \(y\), is a direct variation. When a tour guide works for \(15 \mathrm{hr}\), the cost is \(\$ 1125\). a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the cost to hire a tour guide for \(8 \mathrm{hr}\). d. What does \(k\) represent in this equation?

Step-by-Step Solution

Answer

a. \( k = 75 \, \text{\textdollar/hr} \). b. \( y = 75x \). c. \( y = 600 \, \text{\textdollar} \) for 8 hrs. d. \( k \) represents the cost per hour.

Step 1: Understand the concept of direct variation

In a direct variation relationship, two variables are related in such a way that the ratio of their values is constant. The general form of a direct variation equation is given by: \[ y = kx \] where \( k \) is the constant of proportionality.

Step 2: Identify the given values

From the problem, the given values are: \( x = 15 \, \text{hr} \) \( y = 1125 \, \text{\textdollar} \)

Step 3: Find the constant of proportionality, \( k \)

Using the formula for direct variation: \[ y = kx \] Substitute the given values: \[ 1125 = k \times 15 \] Solve for \( k \): \[ k = \frac{1125}{15} = 75 \, \text{\textdollar/ hr} \]

Step 4: Write the direct variation equation

Now that \( k \) is found, substitute \( k \) into the general form of the direct variation equation: \[ y = 75x \]

Step 5: Find the cost to hire a tour guide for 8 hours

Use the equation \( y = 75x \) and substitute \( x = 8 \): \[ y = 75 \times 8 \] Calculate \( y \): \[ y = 600 \, \text{\textdollar} \]

Step 6: Interpret the constant of proportionality, \( k \)

The constant of proportionality, \( k \), represents the cost per hour to hire the tour guide. In this case, \( k = 75 \, \text{\textdollar/ hr} \) means it costs 75 dollars per hour to hire the tour guide.

Key Concepts

constant of proportionality

The constant of proportionality, often represented by the letter \( k \), is a key concept in direct variation equations. This constant indicates how two variables are related by a fixed ratio. In simpler terms, it's the amount that one variable changes for every unit change in the other variable. For example, in our exercise:

• When a tour guide works for 15 hours, the cost is $1125.
• The constant of proportionality \( k \) can be found using the equation \( y = kx \).

By substituting the given values:
\[ 1125 = k \times 15 \] We solve for \( k \) as follows: \[ k = \frac{1125}{15} = 75 \text{ dollars/hour} \] Thus, \( k \) represents the cost to hire a tour guide per hour, which is 75 dollars for this particular guide.

direct variation equation

A direct variation equation shows the relationship between two variables with a constant ratio. The general form is: \[ y = kx \] Here, \( y \) depends directly on \( x \), and \( k \) is the constant of proportionality. In our example:

• We found that \( k = 75 \) dollars per hour.
• Thus, the direct variation equation becomes \( y = 75x \).

This equation is straightforward.
For any number of hours \( x \), you can calculate the total cost \( y \) by multiplying \( x \) by 75. This keeps math easy and predictable.

solving for variables

Solving for variables in direct variation problems involves using the given direct variation equation. Let's use our equation \( y = 75x \) to see this in action.
To find the cost for different hours worked:

• Plug the number of hours into the equation.
• Multiply by the constant of proportionality.

For example, to find the cost to hire a tour guide for 8 hours:
Substitute \( x = 8 \) into the equation: \[ y = 75 \times 8 \] Calculate \( y \): \[ y = 600 \text{ dollars} \] This shows that hiring the tour guide for 8 hours costs 600 dollars.
Knowing how to solve for variables helps you make quick cost estimates for any number of hours you need.