To find the length of \( b \) (the side opposite angle \( B \)), we can use the formula for the area of triangle \( ABC \):
\[ \text{Area} = \frac{1}{2} a b \sin(C) \]
Where:
- \( a \) is the length of side \( a \) (21 mm),
- \( b \) is the length of side \( b \) (opposite to angle \( B \)),
- \( C \) is the angle at vertex \( C \) (57 degrees),
- and the Area is given as 528.3 mm².
Substituting the known values into the area formula:
\[ 528.3 = \frac{1}{2} \cdot 21 \cdot b \cdot \sin(57^\circ) \]
Now, we can solve for \( b \).
- Calculate \( \sin(57^\circ) \):
Using a calculator, we find:
\[ \sin(57^\circ) \approx 0.8387 \]
- Substitute \( \sin(57^\circ) \) back into the equation:
\[ 528.3 = \frac{1}{2} \cdot 21 \cdot b \cdot 0.8387 \]
- Simplify the equation:
\[ 528.3 = 10.5 \cdot b \cdot 0.8387 \]
Now calculate \( 10.5 \cdot 0.8387 \):
\[ 10.5 \cdot 0.8387 \approx 8.8024 \]
- The equation now looks like this:
\[ 528.3 = 8.8024 \cdot b \]
- Solve for \( b \):
\[ b = \frac{528.3}{8.8024} \]
Calculating that:
\[ b \approx 59.9 \]
So, rounding to the nearest tenth, the length of \( b \) is
\[ \boxed{59.9} \text{ mm} \]
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