Problem 14 - Algebra 2 and Trigonometry

Question

When Brian does odd jobs, he charges ten dollars an hour plus two dollars for transportation. When he works for Mr. Atkins, he receives a 15\(\%\) tip based on his wages for the number of hours that he works but not on the cost of transportation. a. Find \(c(x),\) the amount Brian charges for \(x\) hours of work. b. Find \(t(x),\) Brian's tip when he works for \(x\) hours. c. Find \(c(x),\) Brian's earnings when he works for Mr. Atkins for \(x\) hours, if \(c(x)=c(x)+t(x)\) d. Find e( 3 ), Brian's earnings when he works for Mr. Atkins for 3 hours.

Step-by-Step Solution

Answer

Brian charges \(10x + 2\), tips \(1.5x\), and earns \(11.5x + 2\) with Mr. Atkins. For 3 hours, he earns $36.50.

Step 1: Define the hourly charge function, c(x)

Brian charges \(10 per hour plus \)2 for transportation. Therefore, for \(x\) hours of work, his total charge is \(10x + 2\). Thus, the function \(c(x)\) is given by:\[c(x) = 10x + 2\]

Step 2: Calculate the tip function, t(x)

The tip Brian receives is 15\(\%\) of the wages he earns from working for \(x\) hours. Wages exclude the transportation charge, so it is \(10x\). Therefore, the tip \(t(x)\) is:\[t(x) = 0.15 \times 10x = 1.5x\]

Step 3: Determine total earnings function for Mr. Atkins, e(x)

When Brian works for Mr. Atkins, his earnings \(e(x)\) include both the charge \(c(x)\) and the tip \(t(x)\). Thus, we have:\[e(x) = c(x) + t(x) = (10x + 2) + 1.5x = 11.5x + 2\]

Step 4: Calculate e(3) for 3 hours of work

Plug in \(x = 3\) into the formula for \(e(x)\) to find Brian's earnings when working for Mr. Atkins for 3 hours.\[e(3) = 11.5 \times 3 + 2 = 34.5 + 2 = 36.5\]

Key Concepts

Linear Functions

Linear functions are equations that describe a straight line when plotted on a graph. They have the general formula \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. These functions are called 'linear' because they produce a 'line'. In the problem, Brian's charges are represented by a linear function because the relationship between hours worked and his total charges is direct and consistent.
The function for Brian's charge is \(c(x) = 10x + 2\). Here, \(10x\) represents the cost per hour, which means for each additional hour, the charge increases by \\(10. The \(2\) is the fixed transportation cost, representing the y-intercept. This means that even if Brian works for zero hours, there's still a cost of \\)2 due to transportation.
Understanding linear functions is crucial because it allows you to predict outcomes from input values, efficiently calculate expenses, and understand consistent rates of change, like hourly wages.

Word Problems

Word problems are real-world scenarios presented in mathematical terms. They require you to translate situations into mathematical expressions or equations. In this exercise, the word problem involves Brian's working charges and tips. These types of problems help develop problem-solving skills and an understanding of how math applies to real-world situations.
To tackle word problems, follow these guidelines:

• Read the problem carefully.
• Identify what is being asked.
• Determine the information given and how it relates to what is asked.
• Translate the problem into one or more equations.
• Solve the equations step by step.

The key is to break down the text into manageable mathematical parts. For instance, recognizing that Brian's charge of \\(10 per hour plus \\)2 for transport can be expressed as a linear equation is essential to finding the solution.

Substitution Method

The substitution method is a technique used to solve systems of equations. It involves solving one equation for one variable and then substituting that expression into another equation. This reduces the number of unknowns, making it simpler to solve the problem. In the exercise, substitution helps to find Brian's total earnings by substituting the values into the earnings function.
For example, the equation \(e(x) = c(x) + t(x)\) combines both the charge and the tip functions. Substituting the known linear equations \(c(x) = 10x + 2\) and \(t(x) = 1.5x\) into the earnings equation gives \(e(x) = (10x + 2) + 1.5x\), which simplifies to \(11.5x + 2\).
By substituting a specific value like \(x = 3\) into this equation, you can easily calculate the earnings for that number of working hours. This method is particularly useful when dealing with multiple steps in a problem or when handling various functions that need to be combined.