To find the volume of the gold filling, we need to calculate the volume of the cylindrical part and the volume of the hemispherical top separately, and then add them together.
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Volume of the cylinder: The volume \( V_c \) of a cylinder is given by the formula:
\[ V_c = \pi r^2 h \]
where:
- \( r = 1 \) mm (radius of the cylinder)
- \( h = 3 \) mm (height of the cylinder)
Plugging in the values:
\[ V_c = 3.14 \cdot (1^2) \cdot 3 = 3.14 \cdot 1 \cdot 3 = 9.42 \text{ mm}^3 \]
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Volume of the hemisphere: The volume \( V_h \) of a hemisphere is given by the formula:
\[ V_h = \frac{2}{3} \pi r^3 \]
where \( r = 1 \) mm (radius of the hemisphere).
Plugging in the values:
\[ V_h = \frac{2}{3} \cdot 3.14 \cdot (1^3) = \frac{2}{3} \cdot 3.14 \cdot 1 = \frac{6.28}{3} \approx 2.0933 \text{ mm}^3 \]
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Total Volume of the Filling: Now, we add the volumes of the cylinder and the hemisphere:
\[ V_{\text{total}} = V_c + V_h = 9.42 + 2.0933 \approx 11.5133 \text{ mm}^3 \]
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Cost of the gold filling: The cost per cubic millimeter is $91. Therefore, the total cost can be calculated as follows:
\[ \text{Cost} = V_{\text{total}} \cdot \text{Cost per mm}^3 = 11.5133 \cdot 91 \]
Calculating the cost:
\[ \text{Cost} \approx 1048.7033 \]
Rounding to the nearest cent, the cost of the filling is approximately:
\[ \text{Cost} \approx 1048.70 \]
So, the cost of the gold filling is $1048.70.
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