To determine the range of values for the third side \( x \) of a triangle with side lengths \( a = 42.7 \) mm and \( b = 38.03 \) mm, we can use the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. Therefore, we have the following inequalities:
- \( a + b > x \)
- \( a + x > b \)
- \( b + x > a \)
Substituting \( a \) and \( b \) into these inequalities:
- \( 42.7 + 38.03 > x \)
- \( 42.7 + x > 38.03 \)
- \( 38.03 + x > 42.7 \)
Let's solve each inequality one by one:
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For \( 42.7 + 38.03 > x \): \[ 80.73 > x \quad \text{or} \quad x < 80.73 \]
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For \( 42.7 + x > 38.03 \): \[ x > 38.03 - 42.7 \quad \text{or} \quad x > -4.67 \quad \text{(This does not limit the value since it's negative)} \]
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For \( 38.03 + x > 42.7 \): \[ x > 42.7 - 38.03 \quad \text{or} \quad x > 4.67 \]
Now, combining the results from the inequalities, we find that:
- From the first inequality, \( x < 80.73 \).
- From the third inequality, \( x > 4.67 \).
Thus, we can conclude that the range of values for \( x \) (the third side of the triangle) is:
\[ 4.67 < x < 80.73 \]
Among the provided options, the correct answer is: 4.67 < x < 80.73.