Problem 57 - Calculus Volume 1

Question

The manager at a skateboard shop pays his workers a monthly salary \(S\) of $$\$ 750$$ plus a commission of $$\$ 8.50$$ for each skateboard they sell. a. Write a function \(y=S(x)\) that models a worker's monthly salary based on the number of skateboards \(x\) he or she sells. b. Find the approximate monthly salary when a worker sells 25,40 , or 55 skateboards. c. Use the INTERSECT feature on a graphing calculator to determine the number of skateboards that must be sold for a worker to earn a monthly income of $$\$ 1400$$. (Hint: Find the intersection of the function and the line \(y=1400\).)

Step-by-Step Solution

Answer

a: \(S(x) = 750 + 8.5x\). b: $962.5, $1090, $1217.5. c: Sell 77 skateboards.

Step 1: Identify the Fixed Salary and Commission

First, identify the fixed base salary and the commission per skateboard sold. In this case, the base salary is \( S = 750 \) dollars, and the commission rate per skateboard is \( 8.5 \) dollars.

Step 2: Write the Function

The monthly salary function, \( S(x) \), can be modeled as the sum of the fixed salary and the commission times the number of skateboards sold. Thus, the function is:\[ S(x) = 750 + 8.5x \] where \( x \) is the number of skateboards sold.

Step 3: Calculate Salary for Specific Quantities

Now, calculate the monthly salary for the given quantities of skateboards sold:- For \( x = 25 \):\[ S(25) = 750 + 8.5(25) = 750 + 212.5 = 962.5 \]- For \( x = 40 \):\[ S(40) = 750 + 8.5(40) = 750 + 340 = 1090 \]- For \( x = 55 \):\[ S(55) = 750 + 8.5(55) = 750 + 467.5 = 1217.5 \]

Step 4: Determine Skateboards for Specific Income Using Intersection

To find the number of skateboards \( x \) that must be sold to earn \( 1400 \), equate \( S(x) \) to \( 1400 \):\[ 750 + 8.5x = 1400 \]Subtract 750 from both sides:\[ 8.5x = 1400 - 750 = 650 \]Divide both sides by 8.5:\[ x = \frac{650}{8.5} = 76.47 \]Since \( x \) must be a whole number, the worker needs to sell 77 skateboards.

Key Concepts

Monthly Salary Calculation

Calculating the monthly salary when there is both a fixed salary and additional earnings through commission is quite straightforward. For a skateboard shop worker, the fixed salary is $750 each month. The additional earnings come from a commission earned for each skateboard sold.
It's important to realize that the total monthly salary will change depending on how many skateboards are sold.
To compute this, we utilize a linear function that efficiently encompasses both elements of the salary. In the example given, the commission is $8.50 per skateboard. So, the total salary can be calculated by taking the fixed salary of $750 and adding the earnings from the skateboards:

• If no skateboards are sold, the salary is just $750.
• If 25 skateboards are sold, the commission is $212.5, making the total salary $962.5.
• If 40 skateboards are sold, the commission is $340, making the total salary $1090.
• If 55 skateboards are sold, the commission is $467.5, making the total salary $1217.5.

By doing this calculation, workers can better predict their monthly earnings.

Commission-Based Salary

A commission-based salary is a unique pay structure where part of the worker's payment depends on their performance. In retail or sales jobs, this is a common practice. Such a system motivates employees to work harder and sell more, as their paycheck reflects their activity.
The key to understanding commission-based salary is identifying what constitutes the commission. In this skateboard shop example, each skateboard sold earns the worker an additional $8.50.
This is added to their base salary of $750, resulting in a dynamic salary system. Commission-based salaries are typically represented by linear equations:

• These equations are made up of a fixed base (such as $750) and a variable part (which is dependent on the quantity of skateboards sold).
• This approach makes it easy to calculate total earnings for different sales values, enhancing clarity and precision.

This method provides clear rewards for increased sales effort, benefiting both the employer and employee.

Graphing Calculator Intersection

Using a graphing calculator to find intersections is a useful way to solve functions visually.
This feature allows us to find the point where two graphs meet, which corresponds to the solution of an equation in many practical scenarios.
For instance, in this skateboard salary exercise, a worker wishes to know how many skateboards must be sold to earn \(1400 in a month.You need to intersect the salary function, modeled by \( y = 750 + 8.5x \), with the line \( y = 1400 \):

• First, plot both equations on the graphing calculator.
• Then, use the "intersect" feature.
• The intersect button will identify the \( x \)-value which results in a \)1400 salary.

Results here show an intersection near the value of 77 skateboards, a result derived by setting up an equation \( 750 + 8.5x = 1400 \) and solving for \( x \). This method provides a clear visual representation and an accurate solution for equations involving linear functions.

Function Modeling

Modeling real-life situations with functions is a powerful mathematical approach that turns practical problems into solvable equations.
This method helps in predicting outcomes based on a set of inputs.
In this case, we create a function to model the monthly salary in a commission-based pay structure. The function \( S(x) = 750 + 8.5x \) is a linear model of the salary:

• The fixed component (\(750) represents the constant base salary.
• The variable component (\)8.5x) indicates the commission based on skateboard sales.
• When graphed, it produces a line, with the slope \( 8.5 \) signifying earnings per skateboard.

This kind of function modeling is widely used in various fields, providing clear and useful predictions from given data. It turns complex scenarios into straightforward calculations, making it easier to analyze and interpret financial and statistical results.