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Sagot :
To determine which equation best models the given data, we should analyze the pattern of the units sold, [tex]\( y \)[/tex], over the months, [tex]\( x \)[/tex]. The equation that best fits the data would ideally predict the number of units sold as closely as possible to the values given in the table.
Let's evaluate the equations provided in the options one by one:
1. Option A: [tex]\( y = 46.4x + 191.7 \)[/tex]
- This equation represents a linear relationship.
- Starting with the initial month [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 46.4 \cdot 1 + 191.7 = 238.1 \][/tex]
- This value is quite different from the actual initial value of 150 units, hence it's not likely the best fit.
2. Option B: [tex]\( y = 148.4 \sqrt{x-1} - 154 \)[/tex]
- This equation introduces a square root function.
- Starting with the initial month [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 148.4 \sqrt{1-1} - 154 = 148.4 \cdot 0 - 154 = -154 \][/tex]
- This doesn't match well with the data, as it gives a negative prediction which is far from the actual value.
3. Option C: [tex]\( y = -46.4x + 191.7 \)[/tex]
- This is another linear equation, however with a negative slope.
- Starting with the initial month [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -46.4 \cdot 1 + 191.7 = 145.3 \][/tex]
- This is quite close to the 150 units sold initially, but let's see further values:
[tex]\[ x = 2 : y = -46.4 \cdot 2 + 191.7 = 98.9 \quad (\text{whereas actual } y = 300) \][/tex]
- The prediction diverges significantly, and thus is not the best fit.
4. Option D: [tex]\( y = 148.4 \sqrt{x-1} + 154 \)[/tex]
- Again considering the square root function but with a different constant.
- Starting with the initial month [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 148.4 \sqrt{1-1} + 154 = 148.4 \cdot 0 + 154 = 154 \][/tex]
- This matches quite well with the initial value, and progressing further:
[tex]\[ x = 2 : y = 148.4 \sqrt{2-1} + 154 = 148.4 \cdot 1 + 154 = 302.4 \quad (\text{compared to actual } y = 300) \][/tex]
[tex]\[ x = 3 : y = 148.4 \sqrt{3-1} + 154 \approx 148.4 \cdot 1.414 + 154 \approx 362.2 \quad (\text{actual } y = 380) \][/tex]
[tex]\[ x = 4 : y = 148.4 \sqrt{4-1} + 154 \approx 148.4 \cdot 1.732 + 154 \approx 411.8 \quad (\text{actual } y = 425) \][/tex]
- These values are close to the actual values observed in the sales data.
Given the analysis, Option D: [tex]\( y = 148.4 \sqrt{x-1} + 154 \)[/tex] provides the closest fit to the data. Therefore, Option D is the correct answer.
Let's evaluate the equations provided in the options one by one:
1. Option A: [tex]\( y = 46.4x + 191.7 \)[/tex]
- This equation represents a linear relationship.
- Starting with the initial month [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 46.4 \cdot 1 + 191.7 = 238.1 \][/tex]
- This value is quite different from the actual initial value of 150 units, hence it's not likely the best fit.
2. Option B: [tex]\( y = 148.4 \sqrt{x-1} - 154 \)[/tex]
- This equation introduces a square root function.
- Starting with the initial month [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 148.4 \sqrt{1-1} - 154 = 148.4 \cdot 0 - 154 = -154 \][/tex]
- This doesn't match well with the data, as it gives a negative prediction which is far from the actual value.
3. Option C: [tex]\( y = -46.4x + 191.7 \)[/tex]
- This is another linear equation, however with a negative slope.
- Starting with the initial month [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -46.4 \cdot 1 + 191.7 = 145.3 \][/tex]
- This is quite close to the 150 units sold initially, but let's see further values:
[tex]\[ x = 2 : y = -46.4 \cdot 2 + 191.7 = 98.9 \quad (\text{whereas actual } y = 300) \][/tex]
- The prediction diverges significantly, and thus is not the best fit.
4. Option D: [tex]\( y = 148.4 \sqrt{x-1} + 154 \)[/tex]
- Again considering the square root function but with a different constant.
- Starting with the initial month [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 148.4 \sqrt{1-1} + 154 = 148.4 \cdot 0 + 154 = 154 \][/tex]
- This matches quite well with the initial value, and progressing further:
[tex]\[ x = 2 : y = 148.4 \sqrt{2-1} + 154 = 148.4 \cdot 1 + 154 = 302.4 \quad (\text{compared to actual } y = 300) \][/tex]
[tex]\[ x = 3 : y = 148.4 \sqrt{3-1} + 154 \approx 148.4 \cdot 1.414 + 154 \approx 362.2 \quad (\text{actual } y = 380) \][/tex]
[tex]\[ x = 4 : y = 148.4 \sqrt{4-1} + 154 \approx 148.4 \cdot 1.732 + 154 \approx 411.8 \quad (\text{actual } y = 425) \][/tex]
- These values are close to the actual values observed in the sales data.
Given the analysis, Option D: [tex]\( y = 148.4 \sqrt{x-1} + 154 \)[/tex] provides the closest fit to the data. Therefore, Option D is the correct answer.
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