A metal bar in the shape of a cuboid with a cross-sectional area of 6 cm2 and a length of 15 cm is melted down and recast into two identical cones. Each cone has a height of 3 cm. Show that the radius

Step 1: Calculate the volume of the cuboid. The volume of a cuboid is given by the cross-sectional area multiplied by its length. V_cuboid = Cross-sectional area × Length V_cuboid = 6 cm^2 × 15 cm V_cuboid = 90 cm^3 Step 2: Calculate the total volume of the two cones. The volume of a single cone is given by the formula V_cone = (1)/(3) r^2 h. Given that the height h = 3 cm. V_cone = (1)/(3) r^2 (3) V_cone = r^2 cm^3 Since two identical cones are formed, the total volume of the cones is: V_total\_cones = 2 × V_cone V_total\_cones = 2 r^2 cm^3 Step 3: Equate the volumes and show the expression for r. Since the metal bar is melted to form the cones without any waste, the volume of the cuboid is equal to the total volume of the two cones. V_cuboid = V_total\_cones 90 = 2 r^2 Now, we solve for r^2: r^2 = (90)/(2) r^2 = (45)/() To find r, we take the square root of both sides: r = sqrt((45)/()) We can simplify sqrt(45) as sqrt(9 × 5) = sqrt(9) × sqrt(5) = 3sqrt(5). r = 3sqrt(5)sqrt() r = 3 sqrt((5)/()) This matches the given expression. Step 4: Calculate the value of r using logarithmic tables. Given = 3.14. r = 3 sqrt((5)/(3.14)) To use logarithmic tables, we find r: r = ( 3 × ((5)/(3.14))^(1)/(2) ) Using logarithm properties, (ab) = a + b and (x^n) = n x: r = 3 + (1)/(2) ((5)/(3.14)) Using another property, ((a)/(b)) = a - b: r = 3 + (1)/(2) ( 5 - 3.14) From logarithmic tables: 3 ≈ 0.4771 5 ≈ 0.6990 3.14 ≈ 0.4969 Substitute these values into the equation for r: r = 0.4771 + (1)/(2) (0.6990 - 0.4969) r = 0.4771 + (1)/(2) (0.2021) r = 0.4771 + 0.10105 r = 0.57815 Now, find the antilog of 0.57815 to get the value of r. Using antilog tables, antilog of 0.57815 ≈ 3.7855. Rounding to the first decimal place: r ≈ 3.8 cm The value of r to the first decimal place is 3.8 cm. 3 done, 2 left today. You're making progress.