1. The problem is to find the volume of the frustum of a cone formed by cutting a cone with a diameter of 48 cm parallel to its base. 2. First, note the total height of the original cone is the sum of the two heights given: $$H = 10 + 15 = 25$$ cm. 3. The diameter of the original cone's base is 48 cm, so its radius is $$R = \frac{48}{2} = 24$$ cm. 4. When the cone is cut 10 cm from the top, a smaller cone is removed. We need to find the radius of the smaller cone formed by this 10 cm height. Since the cones are similar, the radius scales with height: $$r = R \times \frac{\text{smaller cone height}}{\text{total height}} = 24 \times \frac{10}{25} = 9.6 \text{ cm}$$ 5. The frustum has height $$h = 15$$ cm (the remaining height after cutting). 6. The volume of a frustum of a cone is given by the formula: $$V = \frac{1}{3} \pi h \left(R^2 + Rr + r^2\right)$$ 7. Substitute the values: $$V = \frac{1}{3} \pi \times 15 \times \left(24^2 + 24 \times 9.6 + 9.6^2\right)$$ Calculate inside the parentheses: $$24^2 = 576$$ $$24 \times 9.6 = 230.4$$ $$9.6^2 = 92.16$$ Sum: $$576 + 230.4 + 92.16 = 898.56$$ 8. Calculate volume: $$V = \frac{1}{3} \times \pi \times 15 \times 898.56 = 5 \pi \times 898.56 = 4492.8 \pi$$ 9. Approximate numerically: $$V \approx 4492.8 \times 3.1416 = 14103.4 \text{ cm}^3$$ 10. Final answer rounded to 1 decimal place: $$\boxed{14103.4 \text{ cm}^3}$$