Problem 70
Question
ECONOMICS: Does Money Buy Happiness? Several surveys in the United States and Europe have asked people to rate their happiness on a scale of \(3={ }^{\prime \prime}\) very happy," \(2=\) "fairly happy," and \(1={ }^{\prime \prime}\) not too happy," and then tried to correlate the answer with the person's income. For those in one income group (making $$\$ 25,000$$ to $$\$ 55,000$$ ) it was found that their "happiness" was approximately given by \(y=0.065 x-0.613\). Find the reported "happiness" of a person with the following incomes (rounding your answers to one decimal place). a. $$\$ 25,000$$ b. $$\$ 35,000$$ c. $$\$ 45,000$$
Step-by-Step Solution
Verified Answer
a. 1,624.4
b. 2,274.4
c. 2,924.4
1Step 1: Understand the Formula
The formula given provides a linear relationship between income and happiness: \[ y = 0.065x - 0.613 \] where \( y \) is the happiness score, and \( x \) is the income.
2Step 2: Calculate Happiness for $25,000
Substitute \( x = 25,000 \) into the formula. \[ y = 0.065(25,000) - 0.613 \] Calculate the multiplication: \[ y = 1,625 - 0.613 \] Subtract to find the happiness score: \[ y = 1,624.387 \] Round to one decimal place: \[ y = 1,624.4 \]
3Step 3: Calculate Happiness for $35,000
Substitute \( x = 35,000 \) into the formula. \[ y = 0.065(35,000) - 0.613 \] Calculate the multiplication: \[ y = 2,275 - 0.613 \] Subtract to find the happiness score: \[ y = 2,274.387 \] Round to one decimal place: \[ y = 2,274.4 \]
4Step 4: Calculate Happiness for $45,000
Substitute \( x = 45,000 \) into the formula. \[ y = 0.065(45,000) - 0.613 \] Calculate the multiplication: \[ y = 2,925 - 0.613 \] Subtract to find the happiness score: \[ y = 2,924.387 \] Round to one decimal place: \[ y = 2,924.4 \]
Key Concepts
Linear EquationsIncome and HappinessStep-by-Step Solutions
Linear Equations
Linear equations describe a straight-line relationship between two variables. In this case, the equation \( y = 0.065x - 0.613 \) links income to happiness.
Here are some key components to understand:
Simply substitute the desired income into the equation to find out the corresponding happiness level.
Here are some key components to understand:
- Variables \(x\) and \(y\): \(x\) represents income, and \(y\) represents happiness.
- Slope: The coefficient 0.065 in front of \(x\) is the slope of the line. It indicates that for each increase of 1 unit in income, the happiness score increases by 0.065 units.
- Intercept: The term \(-0.613\) is the y-intercept, meaning it is the happiness score when the income is zero.
Simply substitute the desired income into the equation to find out the corresponding happiness level.
Income and Happiness
There is a long-standing debate on whether money can buy happiness. This linear relationship attempts to quantify how income affects happiness.
The equation \( y = 0.065x - 0.613 \) suggests that, within the supplied income range of \(25,000 to \)55,000, happiness slightly increases with income.
However, the equation is just a model that illustrates average trends from surveys:
The equation \( y = 0.065x - 0.613 \) suggests that, within the supplied income range of \(25,000 to \)55,000, happiness slightly increases with income.
However, the equation is just a model that illustrates average trends from surveys:
- **Limited Range**: The equation is applicable only for the specific income range given (\(25,000 to \)55,000).
- **Simplified Representation**: It simplifies the complex concept of happiness into a numeric score.
- **Average Trends**: Individual satisfaction may vary, as the equation is based on average data.
Step-by-Step Solutions
Step-by-step solutions help break down complex problems into manageable parts. When working with the linear equation \( y = 0.065x - 0.613 \), follow these steps:
1. **Identify Variables**: Recognize that \(x\) is income, and \(y\) is happiness.2. **Substitute Income**: Replace \(x\) with the specific income amount you want to analyze. For example, use \(x = 25,000\) to start.3. **Calculate Step by Step**: - First, multiply the income by 0.065. - Next, subtract 0.613 from that result to get the happiness score.4. **Round the Result**: Since happiness is rounded to one decimal place, make sure to adjust the final number accordingly.
Example: With \(x = 25,000\), the steps yield a happiness score of 1,624.4.
This process allows you to easily determine the desired outcome one step at a time, minimizing errors and enhancing understanding.
1. **Identify Variables**: Recognize that \(x\) is income, and \(y\) is happiness.2. **Substitute Income**: Replace \(x\) with the specific income amount you want to analyze. For example, use \(x = 25,000\) to start.3. **Calculate Step by Step**: - First, multiply the income by 0.065. - Next, subtract 0.613 from that result to get the happiness score.4. **Round the Result**: Since happiness is rounded to one decimal place, make sure to adjust the final number accordingly.
Example: With \(x = 25,000\), the steps yield a happiness score of 1,624.4.
This process allows you to easily determine the desired outcome one step at a time, minimizing errors and enhancing understanding.
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