To find out how much shaved ice the larger cup can hold, we first need to determine the volume of the smaller cup and then use the scale factor to find the volume of the larger cup.
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Volume of a Cone Formula: The volume \( V \) of a cone is given by the formula: \[ V = \frac{1}{3} \pi r^2 h \] where \( r \) is the radius and \( h \) is the height of the cone.
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Dimensions of the Smaller Cup: The smaller cup has a height \( h = 4 \) inches and a diameter of \( 3 \) inches. Therefore, the radius \( r \) is: \[ r = \frac{3}{2} = 1.5 \text{ inches} \]
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Volume of the Smaller Cup: Plugging the dimensions into the cone volume formula: \[ V = \frac{1}{3} \pi (1.5)^2 (4) \] First, calculate \( (1.5)^2 \): \[ (1.5)^2 = 2.25 \] Now substitute this value back into the volume equation: \[ V = \frac{1}{3} \pi (2.25) (4) = \frac{1}{3} \pi (9) = 3 \pi \] So, the volume of the smaller cup is \( 3\pi \).
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Scale Factor for the Larger Cup: The larger cup is scaled by a factor of 3. When volumes are involved, the volume scales by the cube of the scale factor. Therefore, the volume of the larger cup is calculated as: \[ \text{Volume of larger cup} = \text{Volume of smaller cup} \times (\text{scale factor})^3 = (3\pi) \times (3^3) \] Calculate \( 3^3 \): \[ 3^3 = 27 \] Therefore: \[ \text{Volume of larger cup} = 3\pi \times 27 = 81\pi \]
The correct answer is therefore 81π.
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