A decorative block is made of two solids — a cube and a hemisphere. The base of the block is a cube with edge 5 cm, and the hemisphere fixed on the top has a diameter of 4.2 cm. Find the total surface area of the block
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Step-by-Step Solution
Step 1: Identify Given Dimensions
First, we identify the given dimensions of the cube and the hemisphere. The edge length of the cube is 5 cm, and the diameter of the hemisphere is 4.2 cm.
Step 2: Calculate Hemisphere Radius
The radius of the hemisphere is half of its diameter. So, we divide the given diameter by 2 to find the radius.
Step 3: Calculate Surface Area of Cube
The total surface area of the block is the sum of the surface area of the cube and the curved surface area of the hemisphere, minus the area of the base of the hemisphere that is covered by the cube. The surface area of a cube is 6a2, and the area of the base of the hemisphere is πr2.
Step 4: Calculate Curved Surface Area of Hemisphere
The curved surface area of a hemisphere is given by the formula 2πr2. This is the part of the hemisphere that is exposed.
Step 5: Calculate Total Surface Area of the Block
The total surface area of the block is the surface area of the cube (excluding the area covered by the hemisphere's base) plus the curved surface area of the hemisphere. This simplifies to 6a2+πr2. We substitute the values of a=5 cm and r=2.1 cm into the formula and calculate the result.