Question
Adam calculates his annual salary (base pay and commission), y, using the model y=0.28x + 38,000, where x represents his total sales for the year. What is the y-intercept? What is the slope? What does the y-intercept mean in context? What does the slope mean in context? If he made a 45,000 total annual salary, how much was his total salary?
Answers
GPT-4o mini
In the equation \( y = 0.28x + 38,000 \):
- The **y-intercept** is the value of \( y \) when \( x = 0 \). In this case, the y-intercept is 38,000. This means that if Adam has a total sales of \( 0 \) dollars, his annual salary would be 38,000 dollars.
- The **slope** is the coefficient of \( x \), which is \( 0.28 \). This means that for every additional dollar in total sales \( x \), Adam's salary \( y \) increases by 0.28 dollars (or 28 cents).
### Contextual Meaning:
- The **y-intercept of 38,000** indicates his base pay or guaranteed salary that he earns regardless of his sales performance. It represents the minimum amount he will earn in a year even if he doesn't make any sales.
- The **slope of 0.28** indicates the commission rate Adam earns on his sales. For every dollar of sales he makes, he earns an additional 28 cents in commission. This incentivizes him to increase his sales to boost his overall earnings.
### Calculation of Total Salary:
If Adam made a total annual salary of 45,000 dollars, we can determine how much was his total sales \( x \) by substituting \( y \) with 45,000 in the equation:
\[
45,000 = 0.28x + 38,000
\]
To solve for \( x \):
1. Subtract 38,000 from both sides:
\[
45,000 - 38,000 = 0.28x
\]
\[
7,000 = 0.28x
\]
2. Divide both sides by 0.28:
\[
x = \frac{7,000}{0.28} \approx 25,000
\]
So, Adam's total sales \( x \) would be approximately 25,000 dollars.
To summarize:
- **y-intercept**: 38,000 (base salary)
- **slope**: 0.28 (commission rate)
- **If his total annual salary was 45,000, his total sales were approximately 25,000 dollars.**
- The **y-intercept** is the value of \( y \) when \( x = 0 \). In this case, the y-intercept is 38,000. This means that if Adam has a total sales of \( 0 \) dollars, his annual salary would be 38,000 dollars.
- The **slope** is the coefficient of \( x \), which is \( 0.28 \). This means that for every additional dollar in total sales \( x \), Adam's salary \( y \) increases by 0.28 dollars (or 28 cents).
### Contextual Meaning:
- The **y-intercept of 38,000** indicates his base pay or guaranteed salary that he earns regardless of his sales performance. It represents the minimum amount he will earn in a year even if he doesn't make any sales.
- The **slope of 0.28** indicates the commission rate Adam earns on his sales. For every dollar of sales he makes, he earns an additional 28 cents in commission. This incentivizes him to increase his sales to boost his overall earnings.
### Calculation of Total Salary:
If Adam made a total annual salary of 45,000 dollars, we can determine how much was his total sales \( x \) by substituting \( y \) with 45,000 in the equation:
\[
45,000 = 0.28x + 38,000
\]
To solve for \( x \):
1. Subtract 38,000 from both sides:
\[
45,000 - 38,000 = 0.28x
\]
\[
7,000 = 0.28x
\]
2. Divide both sides by 0.28:
\[
x = \frac{7,000}{0.28} \approx 25,000
\]
So, Adam's total sales \( x \) would be approximately 25,000 dollars.
To summarize:
- **y-intercept**: 38,000 (base salary)
- **slope**: 0.28 (commission rate)
- **If his total annual salary was 45,000, his total sales were approximately 25,000 dollars.**