To find the volume of the rectangular prism, we can use the formula for the volume of a rectangular prism:
\[ V = \text{length} \times \text{width} \times \text{height} \]
Substituting in the given dimensions:
\[ V = 6 , \text{inches} \times 4.5 , \text{inches} \times 2 , \text{inches} \]
Calculating this step-by-step:
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First, calculate the area of the base (length × width): \[ 6 , \text{inches} \times 4.5 , \text{inches} = 27 , \text{inches}^2 \]
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Now, multiply by the height: \[ 27 , \text{inches}^2 \times 2 , \text{inches} = 54 , \text{inches}^3 \]
So, the volume of the rectangular prism is \(54 , \text{inches}^3\).
Next, we calculate the volume of the hemisphere that is cut out of the rectangular prism. The formula for the volume of a hemisphere is:
\[ V = \frac{2}{3} \pi r^3 \]
The diameter of the hemisphere is given as 2.5 inches, hence the radius \(r\) is:
\[ r = \frac{2.5}{2} = 1.25 , \text{inches} \]
Now substituting the radius into the hemisphere's volume formula:
\[ V = \frac{2}{3} \pi (1.25)^3 \]
Calculating \( (1.25)^3 \):
\[ (1.25)^3 = 1.953125 \]
Now substituting this into the volume formula:
\[ V = \frac{2}{3} \pi (1.953125) \]
Calculating \( \frac{2}{3} \times 1.953125 \):
\[ \frac{2}{3} \times 1.953125 \approx 1.30208333333 \]
Now multiplying by \(\pi\):
\[ V \approx 1.30208333333 \times 3.14159265359 \approx 4.0921 \]
Rounding this value gives us approximately:
\[ V \approx 4.1 , \text{inches}^3 \]
Now, we find the total volume after the hemisphere is cut out:
\[ \text{Volume after cutting out hemisphere} = 54 , \text{inches}^3 - 4.1 , \text{inches}^3 \]
Calculating this gives:
\[ 54 - 4.1 \approx 49.9 , \text{inches}^3 \]
Thus, the final volume of the mold is approximately \(49.9 , \text{inches}^3\).
The answer is:
49.9 in.³