An air conditioning salesperson receives a base salary of $2850 per month plus a commission. The commission is 2% of the sales up to and including $25,000 for the month and 5% of the sales over $25,000 for the month.
a. Write a piecewise function that relates the salesperson total monthly income based off or his/her sales for the month.
b. Determine the salesperson's monthly income if his/her sales were $43,000 for the month.
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a. To write a piecewise function that relates the salesperson's total monthly income based on their sales for the month, we can split it into two parts:
Let x represent the sales for the month.
If x ≤ $25,000, the commission is 2% of the sales:
Commission = 0.02 * x
If x > $25,000, the commission is 5% of the sales above $25,000:
Commission = 0.05 * (x - $25,000)
Therefore, the total monthly income can be calculated as:
Total monthly income = Base Salary + Commission
Combining the above information, the piecewise function can be written as:
Total monthly income = $2850 + (0.02 * x) if x ≤ $25,000
Total monthly income = $2850 + (0.05 * (x - $25,000)) if x > $25,000
b. To determine the salesperson's monthly income if their sales were $43,000 for the month, we need to plug in this value in the piecewise function.
For x = $43,000:
Total monthly income = $2850 + (0.05 * ($43,000 - $25,000))
Total monthly income = $2850 + (0.05 * $18,000)
Total monthly income = $2850 + $900
Total monthly income = $3750
Therefore, the salesperson's monthly income, if their sales were $43,000 for the month, would be $3750.
To write a piecewise function that relates the salesperson's total monthly income based on their sales for the month, we need to consider two scenarios: when the sales are up to and including $25,000, and when the sales are over $25,000.
Let's first denote the sales for the month as x.
a. When the sales are up to and including $25,000, the commission rate is 2%. This means for sales up to $25,000, the commission earned would be 2% of the sales.
So, the income from sales up to $25,000 is 0.02 * x.
b. When the sales are over $25,000, the commission rate is 5%. This means for sales over $25,000, the commission earned would be 5% of the sales.
So, the income from sales over $25,000 is 0.05 * (x - 25000).
Now, let's write the piecewise function:
Income(x) =
- (base salary) + (0.02 * x) if x <= 25000
- (base salary) + (0.02 * 25000) + (0.05 * (x - 25000)) if x > 25000
In this case, the base salary is $2850 per month.
b. To determine the salesperson's monthly income if their sales were $43,000 for the month, we can substitute x = 43000 into the piecewise function:
Income(43000) =
- 2850 + (0.02 * 43000) if 43000 <= 25000
- 2850 + (0.02 * 25000) + (0.05 * (43000 - 25000)) if 43000 > 25000
Income(43000) =
- 2850 + 860 if 43000 <= 25000
- 2850 + 500 + 900 if 43000 > 25000
Here, since 43000 is greater than 25000, the second part of the piecewise function applies.
Income(43000) =
- 2850 + 500 + 900
Income(43000) =
$-1450
Therefore, if the salesperson's sales were $43,000 for the month, their monthly income would be -$1450. This means they would have a deficit or loss of $1450 for the month.