Problem 57 - Calculus Volume 1

Question

[T] 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

The function is \(y = 750 + 8.50x\). Salaries are \$962.50, \$1090, and \$1217.50 for 25, 40, and 55 skateboards respectively. About 76-77 skateboards are needed to earn \$1400.

Step 1: Define the Function

The worker's monthly salary is composed of a base salary plus commission for each skateboard sold. Thus, the function that models this is:\[y = 750 + 8.50x\]where \(y\) is the total monthly salary and \(x\) is the number of skateboards sold.

Step 2: Calculate Salary for 25 Skateboards

Substitute \(x = 25\) into the function:\[y = 750 + 8.50 \times 25\]Calculate:\[y = 750 + 212.5 = 962.5\]So, if a worker sells 25 skateboards, their monthly salary is \$962.50.

Step 3: Calculate Salary for 40 Skateboards

Substitute \(x = 40\) into the function:\[y = 750 + 8.50 \times 40\]Calculate:\[y = 750 + 340 = 1090\]Therefore, if a worker sells 40 skateboards, their monthly salary is \$1090.

Step 4: Calculate Salary for 55 Skateboards

Substitute \(x = 55\) into the function:\[y = 750 + 8.50 \times 55\]Calculate:\[y = 750 + 467.5 = 1217.5\]Therefore, if a worker sells 55 skateboards, their monthly salary is \$1217.50.

Step 5: Use Graphing Calculator to Find Intersection

To find how many skateboards must be sold to earn a monthly salary of \\(1400, set up the equation:\[1400 = 750 + 8.50x\]Subtract \\)750 from both sides:\[650 = 8.50x\]Solve for \(x\) by dividing both sides by \$8.50:\[x \approx 76.47\]Using the INTERSECT feature on a graphing calculator, graph the function \(y = 750 + 8.50x\) and the line \(y = 1400\), then find their point of intersection to confirm this solution.

Key Concepts

Monthly Salary Calculation

Understanding how monthly salary is calculated can be crucial for workers who earn a combination of a base salary and commission. In this skateboard shop example, the manager provides a fixed base monthly salary of \\(750. However, to incentivize more sales, each worker earns an additional \\)8.50 for every skateboard they sell. This forms a linear function equation to calculate the total monthly salary.So, the formula to calculate the monthly salary \(y\) concerning the number of skateboards sold \(x\) is:\[y = 750 + 8.50x\]This formula easily breaks down the salary into two parts:

• The base salary of \$750, which remains constant.
• The commission per skateboard, which changes according to sales.

By plugging different values of \(x\) into this formula, workers can predict their earnings based on their sales performance for the month.

Commission Structure

A commission structure like the one in our skateboard shop uses each sale to enhance the worker's monthly earnings, promoting better performance through financial incentives. In our problem, each skateboard sold earns the seller \\(8.50 in addition to their base salary.Understanding this structure is vital because:

• It motivates team members to increase sales, directly linking efforts to rewards.
• It supports transparent earnings calculations. Workers can clearly see how each sale contributes to their total income.

This kind of pay structure typically remains consistent, making it easy for workers to set sales goals to reach desired earnings. Workers know they will receive \\)750 plus \(8.5\times\text{number of skateboards sold}\), ensuring reliability and predictability in income.

Graphing Calculator

A graphing calculator is an excellent tool for performing complex calculations and visualizing functions. In our context, it can be used to find out how many skateboards a worker must sell to achieve a specific salary target, such as \$1400.Here's a simplified process:

• First, input the salary function \(y = 750 + 8.50x\) into the calculator.
• Next, enter the salary target as a constant line, here \(y = 1400\).
• Use the "INTERSECT" feature to find where both lines meet on the graph.

At the intersection, the value of \(x\) tells us the number of skateboards that need to be sold. In our example, solving the equation shows around \(76.47\) skateboards are needed, which can visually and numerically confirmations on the graphing calculator.