Surface Area Analysis of a Cuboidal Block Surmounted by a Hemisphere

Problem Statement and Geometric Configuration

The problem involves a cuboidal block, specifically a cube, with a side length given as $7\,cm$ (recorded in the transcript text as "side tem" but utilized as $7\,cm$ in all calculations). On top of this block is placed a hemisphere, a configuration described as the cube being "surmounted by a hemisphere." The two main objectives are to identify the maximum possible diameter that this hemisphere can possess to fit the cube's dimensions and to calculate the total surface area of the resulting composite solid.

Determination of Maximum Diameter and Radius

To ensure the hemisphere fits perfectly on one of the square faces of the cubical block, its base must not exceed the dimensions of that face. Consequently, the greatest diameter ($d$) the hemisphere can have is equal to the edge length ($a$) of the cube. Based on the provided side length:

$\text{Edge of the cube} (a) = 7\,cm$

$\text{Greatest diameter} (d) = 7\,cm$

Once the diameter is established as $7\,cm$, the radius ($r$) is determined by the formula $r = \frac{d}{2}$, resulting in:

$r = \frac{7}{2}\,cm$

Theoretical Derivation of the Total Surface Area

Calculating the total surface area of the combined solid requires a careful accounting of all exposed surfaces while excluding those that are hidden by the junction of the two shapes. The total surface area of the solid is the sum of the total surface area of the cube and the curved surface area of the hemisphere, minus the area where the two shapes meet (the circular base of the hemisphere).

1. The surface area of the six faces of the cube is given by $6a^2$.
2. The curved surface area of the hemisphere is represented by $2\pi r^2$.
3. The area of the circular base of the hemisphere that is no longer exposed on the cube's surface is $\pi r^2$.

The combined formula is:

$\text{Total Surface Area} = 6a^2 + 2\pi r^2 - \pi r^2$

This simplifies to:

$\text{Total Surface Area} = 6a^2 + \pi r^2$

Detailed Step-by-Step Numerical Calculation

Using the values $a = 7\,cm$ and $r = \frac{7}{2}\,cm$, and substituting $\pi = \frac{22}{7}$, the numerical evaluation is performed as follows:

First, calculate the surface area contribution from the cube:

$6a^2 = 6 \times (7)^2 = 6 \times 49 = 294$

Next, calculate the net surface area contribution from the hemisphere:

$\pi r^2 = \frac{22}{7} \times \left(\frac{7}{2}\right)^2$

$\pi r^2 = \frac{22}{7} \times \frac{49}{4}$

By simplifying the fraction (dividing $49$ by $7$ and $22$ by $2$):

$\pi r^2 = \frac{11 \times 7}{2} = \frac{77}{2} = 38.5$

Finally, add the two components together to find the total surface area of the solid:

$\text{Total Surface Area} = 294 + 38.5 = 332.5\,cm^2$

Thus, the total surface area of the solid is $332.5\,cm^2$.