To find the volume of the bucket, we can model it as a frustum of a cone. A frustum is the portion of a cone that lies between two parallel planes cutting it. In this case, the top diameter is 12 cm, the bottom diameter is 8 cm, and the height (depth) is 4 cm. ### Step-by-Step Explanation 1. **Identify the Dimensions**: - Top diameter (D1) = 12 cm, so the radius (R1) = D1/2 = 12/2 = 6 cm. - Bottom diameter (D2) = 8 cm, so the radius (R2) = D2/2 = 8/2 = 4 cm. - Height (h) = 4 cm. 2. **Volume Formula for a Frustum**: The volume \( V \) of a frustum of a cone can be calculated using the formula: \[ V = \frac{1}{3} \pi h (R_1^2 + R_1 R_2 + R_2^2) \] where: - \( R_1 \) is the radius of the top base, - \( R_2 \) is the radius of the bottom base, - \( h \) is the height of the frustum. 3. **Substituting the Values**: Now, we substitute the values we found into the formula: - \( R_1 = 6 \) cm, - \( R_2 = 4 \) cm, - \( h = 4 \) cm. Plugging these into the formula: \[ V = \frac{1}{3} \pi (4) (6^2 + 6 \cdot 4 + 4^2) \] 4. **Calculating the Terms**: - Calculate \( 6^2 = 36 \). - Calculate \( 6 \cdot 4 = 24 \). - Calculate \( 4^2 = 16 \). - Now, add these values together: \[ 36 + 24 + 16 = 76 \] 5. **Final Calculation**: Now substitute back into the volume formula: \[ V = \frac{1}{3} \pi (4) (76) \] \[ V = \frac{1}{3} \pi (304) \] \[ V = \frac{304\pi}{3} \text{ cm}^3 \] ### Conclusion The correct answer is **A. \( \frac{304\pi}{3} \text{ cm}^3 \)**. ### Explanation of Other Options - **Option B: \( 144\pi \text{ cm}^3 \)**: This value does not account for the correct dimensions of the frustum and is likely derived from incorrect calculations or assumptions about the shape. - **Option C: \( 128\pi \text{ cm}^3 \)**: Similar to option B, this value does not match the calculated volume and suggests a misunderstanding of the frustum volume formula. - **Option D: \( 72\pi \text{ cm}^3 \)**: This is significantly lower than the calculated volume and does not reflect the dimensions provided. ### Revision Summary - The volume of a frustum of a cone is calculated using the formula \( V = \frac{1}{3} \pi h (R_1^2 + R_1 R_2 + R_2^2) \). - Ensure to convert diameters to radii before substituting into the formula. - Carefully perform arithmetic operations to avoid errors in calculations. - The final volume for the given bucket dimensions is \( \frac{304\pi}{3} \text{ cm}^3 \).