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

Step 1: Identify Given Dimensions

First, we identify the given dimensions of the cube and the hemisphere. The edge length of the cube is $5 \text{ cm}$, and the diameter of the hemisphere is $4.2 \text{ cm}$.

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 $6a^2$, and the area of the base of the hemisphere is $\pi r^2$.

Step 4: Calculate Curved Surface Area of Hemisphere

The curved surface area of a hemisphere is given by the formula $2\pi r^2$. 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 $6a^2 + \pi r^2$. We substitute the values of $a=5 \text{ cm}$ and $r=2.1 \text{ cm}$ into the formula and calculate the result.

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