Question
This was an exceptionally dry year for portions of the southwestern United States. Monthly precipitation in Phoenix, Arizona, was recorded in the table and is modeled by y = –0.04088x2 + 0.4485x + 1.862.
In what month did Phoenix receive the lowest amount of precipitation?
Month (x)
Precipitation
January
2.27 inches
February
?
March
?
April
?
May
?
June
?
July
?
August
?
September
2.59 inches
October
?
November
?
December
?
Sketch a graph or fill in the table to answer the question.
January
February
November
December
In what month did Phoenix receive the lowest amount of precipitation?
Month (x)
Precipitation
January
2.27 inches
February
?
March
?
April
?
May
?
June
?
July
?
August
?
September
2.59 inches
October
?
November
?
December
?
Sketch a graph or fill in the table to answer the question.
January
February
November
December
Answers
GPT-4o mini
To determine the month in which Phoenix received the lowest amount of precipitation using the given quadratic model \( y = -0.04088x^2 + 0.4485x + 1.862 \), we first need to understand that this equation represents a parabola that opens downwards (due to the negative coefficient on the quadratic term).
### Steps to Find the Lowest Precipitation:
1. **Identify the Vertex of the Parabola:**
Since the parabola opens downwards, the vertex will represent the maximum value of \( y \) (precipitation). The x-value of the vertex can be found using the formula:
\[
x = -\frac{b}{2a}
\]
where \( a = -0.04088 \) and \( b = 0.4485 \).
2. **Calculating the x-value of the Vertex:**
\[
x = -\frac{0.4485}{2 \times -0.04088} = \frac{0.4485}{0.08176} \approx 5.48
\]
This value (approximately 5.48) suggests that the month with the highest precipitation occurs sometime in June (where \( x = 6 \) corresponds to June).
3. **Evaluate Precipitation for Each Month:**
We will calculate the precipitation for each month of the year using the function for \( x \) from 1 to 12 (representing January through December).
- **January (x = 1)**:
\[
y = -0.04088(1)^2 + 0.4485(1) + 1.862 \approx 2.26962
\]
- **February (x = 2)**:
\[
y = -0.04088(2)^2 + 0.4485(2) + 1.862 \approx 2.20
\]
- **March (x = 3)**:
\[
y = -0.04088(3)^2 + 0.4485(3) + 1.862 \approx 2.10
\]
- **April (x = 4)**:
\[
y = -0.04088(4)^2 + 0.4485(4) + 1.862 \approx 2.00
\]
- **May (x = 5)**:
\[
y = -0.04088(5)^2 + 0.4485(5) + 1.862 \approx 1.86
\]
- **June (x = 6)**:
\[
y = -0.04088(6)^2 + 0.4485(6) + 1.862 \approx 1.76
\]
- **July (x = 7)**:
\[
y = -0.04088(7)^2 + 0.4485(7) + 1.862 \approx 1.73
\]
- **August (x = 8)**:
\[
y = -0.04088(8)^2 + 0.4485(8) + 1.862 \approx 1.70
\]
- **September (x = 9)**:
\[
y = -0.04088(9)^2 + 0.4485(9) + 1.862 \approx 2.59 \text{ (Given)}
\]
- **October (x = 10)**:
\[
y = -0.04088(10)^2 + 0.4485(10) + 1.862 \approx 2.00
\]
- **November (x = 11)**:
\[
y = -0.04088(11)^2 + 0.4485(11) + 1.862 \approx 1.70
\]
- **December (x = 12)**:
\[
y = -0.04088(12)^2 + 0.4485(12) + 1.862 \approx 1.50
\]
### Table of Precipitation:
| Month | Precipitation (inches) |
|------------|-------------------------|
| January | 2.27 |
| February | 2.20 |
| March | 2.10 |
| April | 2.00 |
| May | 1.86 |
| June | 1.76 |
| July | 1.73 |
| August | 1.70 |
| September | 2.59 |
| October | 2.00 |
| November | 1.70 |
| December | 1.50 |
### Conclusion:
Based on the calculations from the model, the month in which Phoenix received the lowest amount of precipitation is **December** with approximately **1.50 inches**.
### Steps to Find the Lowest Precipitation:
1. **Identify the Vertex of the Parabola:**
Since the parabola opens downwards, the vertex will represent the maximum value of \( y \) (precipitation). The x-value of the vertex can be found using the formula:
\[
x = -\frac{b}{2a}
\]
where \( a = -0.04088 \) and \( b = 0.4485 \).
2. **Calculating the x-value of the Vertex:**
\[
x = -\frac{0.4485}{2 \times -0.04088} = \frac{0.4485}{0.08176} \approx 5.48
\]
This value (approximately 5.48) suggests that the month with the highest precipitation occurs sometime in June (where \( x = 6 \) corresponds to June).
3. **Evaluate Precipitation for Each Month:**
We will calculate the precipitation for each month of the year using the function for \( x \) from 1 to 12 (representing January through December).
- **January (x = 1)**:
\[
y = -0.04088(1)^2 + 0.4485(1) + 1.862 \approx 2.26962
\]
- **February (x = 2)**:
\[
y = -0.04088(2)^2 + 0.4485(2) + 1.862 \approx 2.20
\]
- **March (x = 3)**:
\[
y = -0.04088(3)^2 + 0.4485(3) + 1.862 \approx 2.10
\]
- **April (x = 4)**:
\[
y = -0.04088(4)^2 + 0.4485(4) + 1.862 \approx 2.00
\]
- **May (x = 5)**:
\[
y = -0.04088(5)^2 + 0.4485(5) + 1.862 \approx 1.86
\]
- **June (x = 6)**:
\[
y = -0.04088(6)^2 + 0.4485(6) + 1.862 \approx 1.76
\]
- **July (x = 7)**:
\[
y = -0.04088(7)^2 + 0.4485(7) + 1.862 \approx 1.73
\]
- **August (x = 8)**:
\[
y = -0.04088(8)^2 + 0.4485(8) + 1.862 \approx 1.70
\]
- **September (x = 9)**:
\[
y = -0.04088(9)^2 + 0.4485(9) + 1.862 \approx 2.59 \text{ (Given)}
\]
- **October (x = 10)**:
\[
y = -0.04088(10)^2 + 0.4485(10) + 1.862 \approx 2.00
\]
- **November (x = 11)**:
\[
y = -0.04088(11)^2 + 0.4485(11) + 1.862 \approx 1.70
\]
- **December (x = 12)**:
\[
y = -0.04088(12)^2 + 0.4485(12) + 1.862 \approx 1.50
\]
### Table of Precipitation:
| Month | Precipitation (inches) |
|------------|-------------------------|
| January | 2.27 |
| February | 2.20 |
| March | 2.10 |
| April | 2.00 |
| May | 1.86 |
| June | 1.76 |
| July | 1.73 |
| August | 1.70 |
| September | 2.59 |
| October | 2.00 |
| November | 1.70 |
| December | 1.50 |
### Conclusion:
Based on the calculations from the model, the month in which Phoenix received the lowest amount of precipitation is **December** with approximately **1.50 inches**.