A salesperson earns a base salary of $1300 per month, plus a commission equal to 8% of her total sales for the month. Complete the chart by calculating both the salesperson's commission for each monthly sales total and their total monthly earnings.
| Monthly Sales Total | Commission Earned | Total Monthly Earnings |
|---|---|---|
| $0 | $$\square$$ | $$\square$$ |
| $1000 | $$\square$$ | $$\square$$ |
| $2000 | $$\square$$ | $$\square$$ |
| $3000 | $$\square$$ | $$\square$$ |
Is her monthly income a linear function of his monthly sales?
A. Yes, because her monthly income grows by $0.08 for every dollar she sells. This is a constant growth rate.
B. No, because her monthly income grows by a different amount for every dollar she sells. This is not a constant growth rate.
Answer (2)
Calculate the commission earned by multiplying the sales total by 8% (0.08).
Calculate the total monthly earnings by adding the base salary ($1300) to the commission earned.
Determine that the monthly income is a linear function because the commission is a fixed percentage of sales.
Conclude that the monthly income grows by $0.08 for every dollar she sells, indicating a constant growth rate and a linear relationship. The answer is Yes. Y es
Explanation
Understanding the Problem We are given a problem where a salesperson earns a base salary of $1300 per month plus a commission of 8% of her total sales. We need to complete a chart calculating the commission earned and total monthly earnings for different sales totals. We also need to determine if her monthly income is a linear function of her monthly sales.
Calculating Commission and Total Earnings First, we calculate the commission earned for each monthly sales total by multiplying the sales total by 8% (0.08). Then, we calculate the total monthly earnings for each sales total by adding the base salary ($1300) to the commission earned.
Sales Total: 0 F or am o n t h l ys a l es t o t a l o f $0 , t h eco mmi ss i o n e a r n e d i s : 0 × 0.08 = $0 T h e t o t a l m o n t h l ye a r nin g s a re : $1300 + $0 = $1300 $
Sales Total: 1000 F or am o n t h l ys a l es t o t a l o f $1000 , t h eco mmi ss i o n e a r n e d i s : $1000 × 0.08 = $80 T h e t o t a l m o n t h l ye a r nin g s a re : $1300 + $80 = $1380 $
Sales Total: 2000 F or am o n t h l ys a l es t o t a l o f $2000 , t h eco mmi ss i o n e a r n e d i s : $2000 × 0.08 = $160 T h e t o t a l m o n t h l ye a r nin g s a re : $1300 + $160 = $1460 $
Sales Total: 3000 F or am o n t h l ys a l es t o t a l o f $3000 , t h eco mmi ss i o n e a r n e d i s : $3000 × 0.08 = $240 T h e t o t a l m o n t h l ye a r nin g s a re : $1300 + $240 = $1540 $
Completed Chart Now, let's fill in the chart:
Monthly Sales Total
Commission Earned
Total Monthly Earnings
$0
$0
$1300
$1000
$80
$1380
$2000
$160
$1460
$3000
$240
$1540
Linear Function Analysis To determine if her monthly income is a linear function of her monthly sales, we check if the monthly income grows by a constant amount for every dollar she sells. Since the commission is a fixed percentage (8%) of sales, the total monthly earnings increase linearly with sales. For every dollar of sales, her income grows by $0.08. Therefore, her monthly income is a linear function of her monthly sales.
Conclusion The salesperson's monthly income is a linear function of her monthly sales because her monthly income grows by $0.08 for every dollar she sells. This is a constant growth rate.
Examples
Understanding linear functions is crucial in many real-life scenarios, such as calculating earnings based on commission. For example, a real estate agent's income often depends on a base salary plus a commission on sales. If an agent earns a base salary of $2000 per month and a commission of 3% on sales, we can model their monthly income as a linear function: I = 2000 + 0.03 S , where I is the monthly income and S is the total sales. This model helps the agent predict their income based on their sales performance and set realistic financial goals.
The salesperson's commission is calculated as 8% of monthly sales, leading to a linear increase in total earnings based on sales. The completed chart shows total earnings for different sales totals. The monthly income indeed is a linear function of sales, confirming answer A: Yes.
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