To locate the point of intersection between the functions \( y = x + 3 \) and \( y = - (x - 5)^2 + 18 \), we need to set these two equations equal to each other:
\[ x + 3 = - (x - 5)^2 + 18 \]
Now, let's solve for \( x \):
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Rearrange the equation to consolidate terms: \[ (x - 5)^2 + x - 15 = 0 \]
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Expand \( (x - 5)^2 \): \[ (x^2 - 10x + 25) + x - 15 = 0 \] \[ x^2 - 9x + 10 = 0 \]
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Factor the quadratic: \[ (x - 5)(x - 2) = 0 \]
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Solve for \( x \): \[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \] \[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
Now, we find the \( y \)-coordinates for both values of \( x \) using the equation \( y = x + 3 \):
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If \( x = 5 \): \[ y = 5 + 3 = 8 \]
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If \( x = 2 \): \[ y = 2 + 3 = 5 \]
The points of intersection are \( (5, 8) \) and \( (2, 5) \).
Now, we need to consider the precision of the \( y \)-coordinate at the point of intersection.
- The \( y \)-coordinate at the first point of intersection is \( 8 \).
- The \( y \)-coordinate at the second point of intersection is \( 5 \).
Since these coordinates are whole numbers, they can be represented as \( 8.000 \) and \( 5.000 \).
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Rounding to the nearest thousandths place:
- For \( 8 \): \( 8.000 \)
- For \( 5 \): \( 5.000 \)
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Rounding to the nearest hundredths place:
- For \( 8 \): \( 8.00 \)
- For \( 5 \): \( 5.00 \)
Conclusion
Since there are no additional digits after the decimal point in the \( y \)-coordinates, the precision remains the same between rounding to the nearest hundredths and thousandths based on the whole number values.
Thus, the more precise value between the nearest thousandths place and the nearest hundredths place is equal, and there is no difference:
Answer: There is no difference in precision, both yield the same values (8.000 vs. 8.00 and 5.000 vs. 5.00).