A solid sphere is cut into two identical hemispheres. Statement 1: The total volume of two hemispheres is equal to the volume of the original sphere. Statement 2: The total surface area of two hemispheres together is equal to the surface area of the original sphere. Which of the following is valid?

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Statement 1 is true and Statement 2 is false. Explanation: Statement 1: The total volume of two hemispheres is equal to the volume of the original sphere. The volume V of a sphere with radius r is given by: V = $\frac{4}{3}$$\pi$r^3 When a sphere is cut into two hemispheres, each hemisphere will have half the volume of the original sphere. Therefore, the volume of one hemisphere is: `V_("hemisphere") = $\frac{1}{2}$ xx $\frac{4}{3}$$\pi$r^3 = $\frac{2}{3}$$\pi$r^3` Since there are two hemispheres, the total volume of the two hemispheres is: This is equal to the volume of the original sphere. Thus, Statement 1 is true. Statement 2: The total surface area of two hemispheres together is equal to the surface area of the original sphere. The surface area A of a sphere with radius r is given by: A = 4$\pi$r2 When the sphere is cut into two hemispheres, each hemisphere will have: • A curved surface area: 2$\pi$r2 • A flat circular base area: $\pi$r2 The total surface area of one hemisphere is: `V_("hemisphere") = 2$\pi$r^2 + $\pi$r^2 = 3$\pi$r^2` Since there are two hemispheres, the total surface area of the two hemispheres is: 2 $\times$ 3$\pi$r2 = 6$\pi$r2 This is more than the surface area of the original sphere, which is 4$\pi$r2. The additional area comes from the flat circular bases of the hemispheres. Thus, Statement 2 is false.

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