1. **State the problem:** We have two similar cones, Cone A and Cone B. The surface area of Cone A is 108 mm² and the surface area of Cone B is 147 mm². The volume of Cone B is 857.5 mm³. We need to find the volume of Cone A. 2. **Recall the formulas and properties:** For similar solids, the ratio of their surface areas is the square of the ratio of their corresponding linear dimensions. The ratio of their volumes is the cube of the ratio of their corresponding linear dimensions. Let the scale factor (ratio of linear dimensions) from Cone A to Cone B be $k$. Then: $$\frac{\text{Surface Area of A}}{\text{Surface Area of B}} = k^2$$ $$\frac{\text{Volume of A}}{\text{Volume of B}} = k^3$$ 3. **Calculate the scale factor $k$ using surface areas:** $$\frac{108}{147} = k^2$$ Simplify the fraction: $$\frac{108}{147} = \frac{\cancel{108}^{36}}{\cancel{147}^{49}} = \frac{36}{49}$$ So: $$k^2 = \frac{36}{49}$$ 4. **Find $k$ by taking the square root:** $$k = \sqrt{\frac{36}{49}} = \frac{6}{7}$$ 5. **Use $k$ to find the volume of Cone A:** $$\frac{\text{Volume of A}}{857.5} = \left(\frac{6}{7}\right)^3$$ Calculate the cube: $$\left(\frac{6}{7}\right)^3 = \frac{6^3}{7^3} = \frac{216}{343}$$ 6. **Solve for the volume of Cone A:** $$\text{Volume of A} = 857.5 \times \frac{216}{343}$$ Calculate the multiplication: $$\text{Volume of A} = 857.5 \times 0.62944 \approx 539.5$$ **Final answer:** The volume of Cone A is approximately **539.5 mm³**.