a hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. determine the surface area of the remaining solid.
Determine the surface area of the remaining solid after cutting out a hemispherical depression from a cube.
Answer:
To find the surface area of the remaining solid after removing a hemispherical depression from the cube, we need to follow these steps:
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Determine the Side Length of the Cube:
Given that the diameter of the hemisphere is equal to the edge of the cube, and the diameter of a hemisphere is twice the radius, we can conclude that the side length of the cube is equal to the radius of the hemisphere. -
Calculate the Surface Area of the Cube:
The surface area of a cube can be found using the formula:
Surface Area = 6 * (side length)^2 -
Calculate the Curved Surface Area of the Hemisphere:
The curved or lateral surface area of a hemisphere is given by the formula:
Curved Surface Area = 2 * π * (radius)^2 -
Calculate the Total Surface Area of the Remaining Solid:
The total surface area of the remaining solid after cutting the hemispherical depression can be found by subtracting the curved surface area of the hemisphere from the surface area of the cube. -
Substitute the Values and Calculate:
Substitute the value of the side length (equal to the radius) into the formulas for the surface area of the cube and the hemisphere to find their respective areas. Then subtract the hemisphere’s curved surface area from the cube’s surface area to get the total surface area of the remaining solid.
By following these steps and performing the calculations accurately, you can determine the surface area of the remaining solid formed after cutting out a hemispherical depression from a cube.