Here's how to solve the problem: Given: Top radius of frustum ($R$) = 20 cm Bottom radius of frustum ($r$) = 15 cm Height of frustum ($h_f$) = 24 cm a) The Volume of the bucket (frustum) Step 1: Use the formula for the volume of a frustum of a cone. The volume of a frustum ($V_f$) is given by: $$V_f = \frac{1}{3}\pi h_f (R^2 + Rr + r^2)$$ Step 2: Substitute the given values into the formula. $$V_f = \frac{1}{3}\pi (24) (20^2 + (20)(15) + 15^2)$$ $$V_f = 8\pi (400 + 300 + 225)$$ $$V_f = 8\pi (925)$$ $$V_f = 7400\pi$$ The volume of the bucket is $\boxed{\text{7400}\pi \text{ cm}^3}$. b) The Volume of the original cone from which the bucket was obtained. Step 1: Determine the height of the small cone that was cut off ($h_s$) and the total height of the original cone ($H$). Let $H$ be the height of the original cone and $h_s$ be the height of the small cone removed from the top. Using similar triangles, the ratio of radii is equal to the ratio of heights: $$\frac{R}{H} = \frac{r}{h_s}$$ We know $H = h_f + h_s$. Substitute this into the ratio: $$\frac{R}{h_f + h_s} = \frac{r}{h_s}$$ $$R h_s = r (h_f + h_s)$$ $$R h_s = r h_f + r h_s$$ $$R h_s - r h_s = r h_f$$ $$h_s (R - r) = r h_f$$ $$h_s = \frac{r h_f}{R - r}$$ Substitute the values: $R=20$ cm, $r=15$ cm, $h_f=24$ cm. $$h_s = \frac{15 \times 24}{20 - 15}$$ $$h_s = \frac{360}{5}$$ $$h_s = 72 \text{ cm}$$ Step 2: Calculate the total height of the original cone ($H$). $$H = h_f + h_s$$ $$H = 24 + 72$$ $$H = 96 \text{ cm}$$ Step 3: Calculate the volume of the original cone. The volume of a cone ($V_c$) is given by: $$V_c = \frac{1}{3}\pi R^2 H$$ Substitute the values: $R=20$ cm and $H=96$ cm. $$V_c = \frac{1}{3}\pi (20)^2 (96)$$ $$V_c = \frac{1}{3}\pi (400) (96)$$ $$V_c = \pi (400) (32)$$ $$V_c = 12800\pi$$ The volume of the original cone is $\boxed{\text{12800}\pi \text{ cm}^3}$. What's next?