To calculate the total volume of ice cream both inside the ice cream cone and on top (the hemisphere), we will follow these steps:
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Find the volume of the hemisphere on top: The volume \( V \) of a hemisphere is given by the formula: \[ V = \frac{2}{3} \pi r^3 \] We know from the problem that the volume of the hemisphere is 4 in³. Thus, we set up the equation: \[ \frac{2}{3} \pi r^3 = 4 \] Solving for \( r^3 \): \[ r^3 = \frac{4 \times 3}{2\pi} = \frac{12}{2\pi} = \frac{6}{\pi} \] To find \( r \): \[ r = \sqrt[3]{\frac{6}{\pi}} \]
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Determine the dimensions of the ice cream cone: The diameter of the cone is equal to its height, say the height \( h \). If the diameter is \( d \), then the radius \( r \) of the cone is: \[ r = \frac{d}{2} = \frac{h}{2} \] Therefore, we have: \[ d = h \quad \text{and} \quad r = \frac{h}{2} \]
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Calculate the volume of the cone: The volume \( V \) of a cone is given by the formula: \[ V = \frac{1}{3} \pi r^2 h \] Substituting \( r = \frac{h}{2} \): \[ V = \frac{1}{3} \pi \left(\frac{h}{2}\right)^2 h \] This simplifies to: \[ V = \frac{1}{3} \pi \frac{h^2}{4} h = \frac{1}{12} \pi h^3 \]
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Relate the height to the radius of the hemisphere: From step 1, we have: \[ r^3 = \frac{6}{\pi} \] Since \( r = \frac{h}{2} \): \[ \left(\frac{h}{2}\right)^3 = \frac{6}{\pi} \] Simplifying gives: \[ \frac{h^3}{8} = \frac{6}{\pi} \] Rearranging yields: \[ h^3 = \frac{48}{\pi} \] Now, substituting back into the volume of the cone: \[ V_\text{cone} = \frac{1}{12} \pi h^3 = \frac{1}{12} \pi \left(\frac{48}{\pi}\right) = \frac{48}{12} = 4 \text{ in}^3 \]
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Find the total volume of ice cream: Now we add the volume of the hemisphere and the cone: \[ V_\text{total} = V_\text{cone} + V_\text{hemisphere} = 4 + 4 = 8 \text{ in}^3 \]
Thus, the total volume of ice cream in and on top of the cone is \( \boxed{8} \text{ in}^3 \).