To analyze the linear equation \( y = 0.2x + 15 \), we need to understand what each component represents:
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Identifying the terms:
- \(y\) represents the average temperature in degrees Celsius.
- \(x\) represents the number of years since 2000. Therefore, if \(x = 0\), it represents the year 2000, if \(x = 1\), it’s 2001, and so on.
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Slope Analysis:
- The coefficient of \(x\) (which is \(0.2\)) represents the rate of change in temperature per year. Thus, the average increase in temperature each year is 0.2°C, not 1°C.
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Finding temperature in the year 2010:
- For the year 2010, \(x = 2010 - 2000 = 10\).
- Plugging \(10\) into the equation: \[ y = 0.2(10) + 15 = 2 + 15 = 17°C. \]
- Therefore, it is correct that in the year 2010 the temperature was 17°C.
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Predicting temperature in 2030:
- For the year 2030, \(x = 2030 - 2000 = 30\).
- In 2029, \(x = 29\): \[ y_{2029} = 0.2(29) + 15 = 5.8 + 15 = 20.8°C. \]
- In 2030: \[ y_{2030} = 0.2(30) + 15 = 6 + 15 = 21°C. \]
- Thus, the temperature in 2030 will indeed be 0.2°C higher than the temperature in 2029, not 1°C higher.
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Average increase every 5 years:
- The average increase in temperature over 5 years (from \(x\) to \(x + 5\)): \[ \text{Increase} = y(x + 5) - y(x) = [0.2(x + 5) + 15] - [0.2x + 15] = 0.2x + 1 - 0.2x = 1°C. \]
- Thus, the average increase in temperature every 5 years is 1°C.
Summary:
Taking all this into account, the true statements from the options given are:
- True: "In the year 2010 the temperature was 17 °C."
- True: "If the trend continues, the temperature in 2030 will be 0.2°C higher than the temperature in 2029." (Note: this statement as it is written about 1°C is false.)
- True: "The average increase in temperature every 5 years is 1°C."
The incorrect statements are regarding the average annual temperature increase (which is 0.2°C) and the temperature difference between 2030 and 2029 (which is also 0.2°C).
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