This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the calculations for the containers: 6.1 Calculate which container has the largest volume. Give the volume to the nearest litre. Step 1: Convert all dimensions to meters for consistency. For the rectangular container: Length (l) = 80 cm = 0.8 m Width (w) = 80 cm = 0.8 m Height (h) = 2 m For the cylindrical container: Diameter (D) = 1 m Radius (r) = (D)/(2) = 1 m2 = 0.5 m Height (h) = 2 m Step 2: Calculate the volume of the rectangular container. V_rectangular = l × w × h V_rectangular = 0.8 m × 0.8 m × 2 m V_rectangular = 1.28 m^3 Step 3: Convert the volume of the rectangular container to litres. Given 1000 cm^3 = 1 l, and 1 m^3 = (100 cm)^3 = 1,000,000 cm^3. Therefore, 1 m^3 = (1,000,000)/(1000) l = 1000 l. V_rectangular = 1.28 m^3 × 1000 lm^3 V_rectangular = 1280 l Step 4: Calculate the volume of the cylindrical container. V_cylindrical = r^2 h V_cylindrical = × (0.5 m)^2 × 2 m V_cylindrical = × 0.25 m^2 × 2 m V_cylindrical = 0.5 m^3 Using ≈ 3.14159: V_cylindrical ≈ 0.5 × 3.14159 m^3 V_cylindrical ≈ 1.570795 m^3 Step 5: Convert the volume of the cylindrical container to litres and round to the nearest litre. V_cylindrical ≈ 1.570795 m^3 × 1000 lm^3 V_cylindrical ≈ 1570.795 l V_cylindrical ≈ 1571 l Step 6: Compare the volumes. 1571 l > 1280 l. The cylindrical container has the largest volume. The container with the largest volume is the cylindrical container, with a volume of 1571 l. 6.2 Calculate the surface area of each container. Step 1: Calculate the surface area of the rectangular container. SA_rectangular = 2(lw + lh + wh) SA_rectangular = 2((0.8 m × 0.8 m) + (0.8 m × 2 m) + (0.8 m × 2 m)) SA_rectangular = 2(0.64 m^2 + 1.6 m^2 + 1.6 m^2) SA_rectangular = 2(3.84 m^2) SA_rectangular = 7.68 m^2 Step 2: Calculate the surface area of the cylindrical container. SA_cylindrical = 2 r^2 + 2 rh SA_cylindrical = 2 (0.5 m)^2 + 2 (0.5 m)(2 m) SA_cylindrical = 2 (0.25 m^2) + 2 (1 m^2) SA_cylindrical = 0.5 m^2 + 2 m^2 SA_cylindrical = 2.5 m^2 Using ≈ 3.14159: SA_cylindrical ≈ 2.5 × 3.14159 m^2 SA_cylindrical ≈ 7.853975 m^2 The surface area of the rectangular container is 7.68 m^2. The surface area of the cylindrical container is 7.85 m^2 (rounded to two decimal places). 6.3 Metal sheets, 2 m wide, are used to manufacture these containers. Calculate the length of sheeting required to make each container. Step 1: Calculate the length of sheeting for the rectangular container. Length of sheeting = Total Surface Area / Width of sheet L_rectangular = SA_rectangularwidth of sheet L_rectangular = 7.68 m^22 m L_rectangular = 3.84 m Step 2: Calculate the length of sheeting for the cylindrical container. L_cylindrical = SA_cylindricalwidth of sheet L_cylindrical = 7.853975 m^22 m L_cylindrical ≈ 3.9269875 m L_cylindrical ≈ 3.93 m The length of sheeting required for the rectangular container is 3.84 m. The length of sheeting required for the cylindrical container is 3.93 m (rounded to two decimal places). That's 2 down. 3 left today — send the next one.