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.
Solution and Explanation
Diameter of hemisphere = Edge of cube =\(l\)
Radius of hemisphere =\(\frac{ l}{2}\)
Total surface area of solid = Surface area of cubical part + CSA of hemispherical part − Area of base of hemispherical part
\(= 6 (\text{edge})^2+2\pi r^2-\pi r^2\)
\(= 6l^2 + \pi r^2\)
\(= 6l^2+\pi(\frac{l}{2})^2\)
\(= 6l^2+\frac{\pi l^2}{4}\)
\(=\frac{ l^2}{4}(24+\pi )\) sq. units
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Concepts Used:
Surface Area and Volume
Surface area and volume are two important concepts in geometry that are used to measure the size and shape of three-dimensional objects.
Surface area is the measure of the total area that the surface of an object covers. It is expressed in square units, such as square meters or square inches. To calculate the surface area of an object, we find the area of each face or surface and add them together. For example, the surface area of a cube is equal to six times the area of one of its faces.
Volume, on the other hand, is the measure of the amount of space that an object takes up. It is expressed in cubic units, such as cubic meters or cubic feet. To calculate the volume of an object, we measure the length, width, and height of the object and multiply these three dimensions together. For example, the volume of a cube is equal to the length of one of its edges cubed.
Surface area and volume are important in many fields, such as architecture, engineering, and manufacturing. For example, surface area is used to calculate the amount of material needed to cover an object, while volume is used to determine the amount of space that a container can hold. Understanding surface area and volume is also important in calculus and physics, where they are used to model the behavior of objects in three-dimensional space.