A solid object is made of a cone on a hemisphere. The cone's slant height is 10 cm and its curved surface area is 60π cm². The hemisphere's radius is the same as the cone's radius. This object is plac

You're on a roll — Here's the solution for the geometry problem: Given: Cylinder height (H) = 24 cm Cylinder radius (R) = 10 cm Initial water height (h_w) = 14.4 cm Solid object: a cone on a hemisphere. Cone slant height (l) = 10 cm Cone curved surface area (CSA_c) = 60 cm^2 Hemisphere radius (r_h) = Cone radius (r_c) Take = 3.14 Part 1: Find the rise in the water level. Step 1: Find the radius of the cone (r_c). The curved surface area of a cone is given by CSA_c = r_c l. 60 cm^2 = × r_c × 10 cm 60 = 10 r_c r_c = (60)/(10) cm r_c = 6 cm Step 2: Find the height of the cone (h_c). Using the Pythagorean theorem for the cone: l^2 = r_c^2 + h_c^2. (10 cm)^2 = (6 cm)^2 + h_c^2 100 cm^2 = 36 cm^2 + h_c^2 h_c^2 = 100 cm^2 - 36 cm^2 h_c^2 = 64 cm^2 h_c = sqrt(64 cm)^2 h_c = 8 cm Step 3: Determine the radius of the hemisphere (r_h). The hemisphere has the same radius as the cone. r_h = r_c = 6 cm Step 4: Calculate the volume of the cone (V_c). The volume of a cone is given by V_c = (1)/(3) r_c^2 h_c. V_c = (1)/(3) × 3.14 × (6 cm)^2 × 8 cm V_c = (1)/(3) × 3.14 × 36 cm^2 × 8 cm V_c = 3.14 × 12 × 8 cm^3 V_c = 3.14 × 96 cm^3 V_c = 301.44 cm^3 Step 5: Calculate the volume of the hemisphere (V_h). The volume of a hemisphere is given by V_h = (2)/(3) r_h^3. V_h = (2)/(3) × 3.14 × (6 cm)^3 V_h = (2)/(3) × 3.14 × 216 cm^3 V_h = 2 × 3.14 × 72 cm^3 V_h = 3.14 × 144 cm^3 V_h = 452.16 cm^3 Step 6: Calculate the total volume of the solid object (V_s). V_s = V_c + V_h V_s = 301.44 cm^3 + 452.16 cm^3 V_s = 753.60 cm^3 Step 7: Calculate the rise in water level ( h). The volume of water displaced is equal to the volume of the solid object. This displaced volume forms a cylinder with radius R and height h. Volume of displaced water = R^2 h. 753.60 cm^3 = 3.14 × (10 cm)^2 × h 753.60 cm^3 = 3.14 × 100 cm^2 × h 753.60 cm^3 = 314 cm^2 × h h = 753.60 cm^3314 cm^2 h = 2.4 cm The rise in the water level is 2.4 cm. Part 2: Find the amount of water still to be filled to the brim. Step 1: Calculate the new water level (h_new). h_new = h_w + h h_new = 14.4 cm + 2.4 cm h_new = 16.8 cm (Note: The total height of the solid is h_c + r_h = 8 cm + 6 cm = 14 cm. Since 16.8 cm > 14 cm, the solid is indeed completely immersed.) Step 2: Calculate the remaining height to be filled. h_fill = H - h_new h_fill = 24 cm - 16.8 cm h_fill = 7.2 cm Step 3: Calculate the volume of water still to be filled (V_fill). V_fill = R^2 h_fill V_fill = 3.14 × (10