6. A farm corn storage tower has two parts, a big cylinder and a semi-sphere on the top. The radius of the base and the semi-sphere is 10 feet, the height of the cylinder part is 60 feet. What is the surface area and the volume of the whole tower? ( 5 marks)

To find the surface area and volume of the corn storage tower, we need to calculate the volume and surface area of both the cylindrical part and the semi-spherical part separately, and then combine the results. ### Given: - Radius of the base and semi-sphere, \( r = 10 \) feet - Height of the cylindrical part, \( h = 60 \) feet ### 1. Volume Calculation **Volume of the Cylinder:** The volume \( V_c \) of a cylinder is given by the formula: \[ V_c = \pi r^2 h \] Substituting the values: \[ V_c = \pi (10)^2 (60) = \pi (100)(60) = 6000\pi \text{ cubic feet} \] **Volume of the Semi-Sphere:** The volume \( V_s \) of a sphere is given by the formula: \[ V_s = \frac{4}{3} \pi r^3 \] Since we only need the volume of a semi-sphere, we take half of the sphere's volume: \[ V_{semi-sphere} = \frac{1}{2} \left(\frac{4}{3} \pi r^3\right) = \frac{2}{3} \pi r^3 \] Substituting the radius: \[ V_{semi-sphere} = \frac{2}{3} \pi (10)^3 = \frac{2}{3} \pi (1000) = \frac{2000}{3} \pi \text{ cubic feet} \] **Total Volume:** \[ V_{total} = V_c + V_{semi-sphere} = 6000\pi + \frac{2000}{3}\pi \] To combine these, we need a common denominator: \[ 6000\pi = \frac{18000}{3}\pi \] Thus, \[ V_{total} = \frac{18000}{3}\pi + \frac{2000}{3}\pi = \frac{20000}{3}\pi \text{ cubic feet} \] ### 2. Surface Area Calculation **Surface Area of the Cylinder:** The surface area \( A_c \) of a cylinder (excluding the top) is given by: \[ A_c = 2\pi r h + \pi r^2 \] Where \( 2\pi r h \) is the lateral surface area and \( \pi r^2 \) is the area of the base. However, since the top of the cylinder is covered by the semi-sphere, we only consider the lateral area and the base area: \[ A_c = 2\pi r h + \pi r^2 \] Substituting the values: \[ A_c = 2\pi (10)(60) + \pi (10)^2 = 1200\pi + 100\pi = 1300\pi \text{ square feet} \] **Surface Area of the Semi-Sphere:** The surface area \( A_s \) of a sphere is given by: \[ A_s = 4\pi r^2 \] For a semi-sphere, we take half of this area and add the base area (which is a circle): \[ A_{semi-sphere} = \frac{1}{2}(4\pi r^2) + \pi r^2 = 2\pi r^2 + \pi r^2 = 3\pi r^2 \] Substituting the radius: \[ A_{semi-sphere} = 3\pi (10)^2 = 3\pi (100) = 300\pi \text{ square feet} \] **Total Surface Area:** \[ A_{total} = A_c + A_{semi-sphere} = 1300\pi + 300\pi = 1600\pi \text{ square feet} \] ### Final Results - **Total Volume:** \( \frac{20000}{3}\pi \) cubic feet - **Total Surface Area:** \( 1600\pi \) square feet These results can be approximated using \( \pi \approx 3.14 \) if needed for numerical values.