To find the volume of the gold filling, we need to calculate the volumes of both the cylindrical part and the hemispherical part.
1. Volume of the cylinder:
The formula for the volume of a cylinder is:
\[ V_{\text{cylinder}} = \pi r^2 h \]
where:
- \( r \) is the radius
- \( h \) is the height
Given:
- \( r = 1 \) mm
- \( h = 4 \) mm
\[ V_{\text{cylinder}} = \pi (1, \text{mm})^2 (4, \text{mm}) = \pi (1) (4) = 4\pi, \text{mm}^3 \]
Using \( \pi \approx 3.14 \):
\[ V_{\text{cylinder}} = 4 \times 3.14 = 12.56, \text{mm}^3 \]
2. Volume of the hemisphere:
The formula for the volume of a hemisphere is:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \]
Given:
- \( r = 1 \) mm
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (1, \text{mm})^3 = \frac{2}{3} \pi (1) = \frac{2\pi}{3}, \text{mm}^3 \]
Using \( \pi \approx 3.14 \):
\[ V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \approx 2.0933, \text{mm}^3 \]
3. Total Volume:
Now, add the volume of the cylinder and the hemisphere to get the total volume of the filling:
\[ V_{\text{total}} = V_{\text{cylinder}} + V_{\text{hemisphere}} = 12.56, \text{mm}^3 + 2.0933, \text{mm}^3 \approx 14.6533, \text{mm}^3 \]
4. Cost of the gold:
Now, to calculate the cost:
The cost of dental gold is \( 94 \) dollars per cubic millimeter.
\[ \text{Cost} = V_{\text{total}} \times \text{cost per mm}^3 \]
\[ \text{Cost} = 14.6533, \text{mm}^3 \times 94, \text{dollars/mm}^3 \approx 1378.88, \text{dollars} \]
Final Answers:
- Volume of the filling: \( \approx 14.6533 , \text{mm}^3 \)
- Cost of the gold for the filling: \( \approx 1378.88 , \text{dollars} \)
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