91Ó°ÊÓ

Bob's Pest Removal Service specializes in removing wild creatures (skunks, bats, reptiles, etc.) from private homes. He charges \(\$ 70\) to go to a house plus \(\$ 20\) per hour for his services. Let \(y\) be the total amount (in dollars) paid by a household using Bob's services and \(x\) the number of hours Bob spends capturing and removing the animal(s). The equation for the relationship between \(x\) and \(y\) is $$ y=70+20 x $$ a. Bob spent 3 hours removing a coyote from under Alice's house. How much will he be paid? b. Suppose nine persons called Bob for assistance during a week. Strangely enough, each of these jobs required exactly 3 hours. Will each of these clients pay Bob the same amount, or do you expect each one to pay a different amount? Explain. c. Is the relationship between \(x\) and \(y\) exact or nonexact?

Step by step solution

01

Calculate the total payment for Alice

First, substitute \(x = 3\) into the equation to find the total amount Alice has to pay. Use the provided formula \(y = 70 + 20x\). Thus, we get \(y = 70 + 20 * 3\). Calculate this to find out the amount Alice will pay.

02

Analyze whether the payment amount will be the same for all clients

The equation does not include any individual characteristics of the clients. It only considers the number of hours of service. If all the jobs take exactly 3 hours, substituting \(x = 3\) into the equation will provide the same result for all the nine clients.

03

Assess the type of the relationship

Observe that for every increment in \(x\) (hours), \(y\) (cost) increases by a constant amount (20 dollars). So this means, the relationship between \(x\) and \(y\) is exact. The cost does not depend on other factors apart from the rate of cost per hour and the number of hours.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cost Calculation

Understanding cost calculation is essential when dealing with linear functions related to real-life scenarios like Bob's Pest Removal Service. In this example, Bob has a fixed fee and an additional fee that depends on the hours he works. The total cost to a customer is calculated using the linear equation \( y = 70 + 20x \), where:

• 70 is the fixed charge for the visit
• 20 is the rate charged per hour
• \( x \) represents the number of hours Bob works

For example, if Bob works for 3 hours, we substitute \( x = 3 \) into the equation: \[ y = 70 + 20 \times 3 \]This gives us:

• \( y = 70 + 60 \)
• \( y = 130 \)

Thus, in this scenario, Alice would pay Bob \( \$130 \). Each customer's cost can be easily calculated using this method.

Exact Relationships

Some relationships in linear functions like Bob's pricing model show exactness. The relationship between the variables \( x \) (hours) and \( y \) (total cost) is direct and consistent. For every additional hour Bob works, the total cost increases by a fixed amount (\\(20). This remains true regardless of other conditions or differences among clients.In such cases, the equation \( y = 70 + 20x \) doesn't vary. It continually provides the same output when you input the same value for \( x \). Let's consider nine customers each requiring 3 hours of service. Since the equation lacks unique client variables other than \( x \), it predictably yields the identical cost: \( \\)130 \) every time \( x = 3 \) is used. This highlights the concept of exactness in linear relationships.

Hourly Rates

Hourly rates are a common component in linear cost functions. They represent the variable part of the costing structure in Bob's Pest Removal Service. In this example, Bob charges \\(20 per hour, meaning that every hour he works increases the total cost by \\)20.With an equation like \( y = 70 + 20x \), each time \( x \) is increased by one (an additional hour), the total cost, \( y \), increases by 20. This makes predicting costs straightforward.- If Bob works 1 hour, the cost is \( 70 + 20 \times 1 = 90 \).- For 2 hours, it's \( 70 + 20 \times 2 = 110 \).- For 3 hours, it's \( 70 + 20 \times 3 = 130 \), and so on.Hourly rates are crucial in services because they allow both companies and clients to easily foresee the expenses involved, ensuring transparency in the billing process.