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Question
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?
Options
Both the statements are true.
Both the statements are false.
Statement 1 is true and Statement 2 is false.
Statement 1 is false and Statement 2 is true.
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Solution
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 = 4/3πr^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") = 1/2 xx 4/3πr^3 = 2/3πr^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 = 4πr2
When the sphere is cut into two hemispheres, each hemisphere will have:
- A curved surface area: 2πr2
- A flat circular base area: πr2
The total surface area of one hemisphere is:
`V_("hemisphere") = 2πr^2 + πr^2 = 3πr^2`
Since there are two hemispheres, the total surface area of the two hemispheres is:
2 × 3πr2 = 6πr2
This is more than the surface area of the original sphere, which is 4πr2.
The additional area comes from the flat circular bases of the hemispheres.
Thus, Statement 2 is false.
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