Question
The cost of a school banquet is $95 plus $15 for each person attending.
What is the slope or rate of change?
What is the meaning of the slope in context?
In 1980 the average price of a home in Brainerd County was $97,000. By 1986 the average price of a home was $109,000.
What is the slope or rate of change in the scenario?
What does the slope mean in context?
What is the slope or rate of change?
What is the meaning of the slope in context?
In 1980 the average price of a home in Brainerd County was $97,000. By 1986 the average price of a home was $109,000.
What is the slope or rate of change in the scenario?
What does the slope mean in context?
Answers
GPT-4o mini
### Banquet Cost
1. **Slope or Rate of Change:**
The cost function for the banquet can be expressed as:
\[
C(x) = 95 + 15x
\]
where:
- \(C(x)\) is the total cost,
- \(x\) is the number of people attending.
The slope (rate of change) is the coefficient of \(x\), which is **15**.
2. **Meaning of the Slope in Context:**
In the context of the school banquet, the slope of 15 indicates that for each additional person attending the banquet, the total cost increases by $15. This reflects the per-person cost associated with the banquet expenses.
### Home Prices in Brainerd County
1. **Slope or Rate of Change:**
To find the slope (rate of change) of home prices from 1980 to 1986, we first note the two data points:
- In 1980: \(y_1 = 97,000\)
- In 1986: \(y_2 = 109,000\)
The years can be treated as \(x_1 = 0\) (1980) and \(x_2 = 6\) (1986). The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{109,000 - 97,000}{6 - 0} = \frac{12,000}{6} = 2,000
\]
Thus, the slope is **$2,000**.
2. **Meaning of the Slope in Context:**
In the context of home prices in Brainerd County, the slope of $2,000 indicates that the average price of a home increased by $2,000 per year from 1980 to 1986. This signifies a steady rise in housing prices during that six-year period.
1. **Slope or Rate of Change:**
The cost function for the banquet can be expressed as:
\[
C(x) = 95 + 15x
\]
where:
- \(C(x)\) is the total cost,
- \(x\) is the number of people attending.
The slope (rate of change) is the coefficient of \(x\), which is **15**.
2. **Meaning of the Slope in Context:**
In the context of the school banquet, the slope of 15 indicates that for each additional person attending the banquet, the total cost increases by $15. This reflects the per-person cost associated with the banquet expenses.
### Home Prices in Brainerd County
1. **Slope or Rate of Change:**
To find the slope (rate of change) of home prices from 1980 to 1986, we first note the two data points:
- In 1980: \(y_1 = 97,000\)
- In 1986: \(y_2 = 109,000\)
The years can be treated as \(x_1 = 0\) (1980) and \(x_2 = 6\) (1986). The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{109,000 - 97,000}{6 - 0} = \frac{12,000}{6} = 2,000
\]
Thus, the slope is **$2,000**.
2. **Meaning of the Slope in Context:**
In the context of home prices in Brainerd County, the slope of $2,000 indicates that the average price of a home increased by $2,000 per year from 1980 to 1986. This signifies a steady rise in housing prices during that six-year period.