To find the total volume of ice cream in the cone and on top, we first need to calculate the volume of the cone. We know that the cone's volume is equal to its height since its diameter is equal to its height, which means its height is 2r (where r is the radius) and the volume formula for a cone is \(V = \frac{1}{3} \pi r^2 h\); so if we let the height be \(h = 2r\), the volume equation becomes \(V = \frac{2}{3} \pi r^3\). Since the volume of the hemisphere is given as 4 in³ (and it has a volume formula of \(V = \frac{2}{3} \pi r^3\)), we can set it equal to 4 to find \(r\) and then use that to find the volume of the cone, then we just add the two volumes together for the total.
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Volume of the hemisphere: \[ V = \frac{2}{3} \pi r^3 = 4 \] Solving for \(r\) gives \(r^3 = 6/\pi\).
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Volume of the cone: Using the same \(r^3\), we put \(r\) back into the cone volume formula: \[ V_{cone} = \frac{2}{3} \pi r^3 = 4 \] Which means the total volume of the ice cream in the cone and on top is: \[ V_{total} = V_{cone} + V_{hemisphere} = 4 + 4 = 8 \text{ in}^3. \] So the total volume of the ice cream in the cone and on top is 8 in³.
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