Question 4

We need to find the greatest diameter of the hemisphere and the total surface area of the solid.

Step 1 — Find the greatest diameter

The hemisphere sits on top of the cube. For the greatest diameter, its base must cover the entire top face. So, the diameter of the hemisphere will be equal to the side of the cube.

Diameter of hemisphere = side of cubical block

$= \mathbf{7 \text{ cm}}$

$\boxed{\text{Greatest diameter} = 7 \text{ cm}}$

Step 2 — Calculate the surface area of the solid

First, let's find the radius of the hemisphere. The radius is half of the diameter:

$r = \frac{\text{Diameter}}{2}$

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

$= \mathbf{3.5 \text{ cm}}$

The hemisphere sits on one face of the cube. So the visible surface is:

• the whole surface of the cube, but with a circle removed where the hemisphere covers it, plus
• the curved surface of the hemisphere.

We use these formulas:

• Surface area of a cube $= 6a^2$ (a cube has 6 equal square faces, each of area $a^2$)
• Curved surface area (CSA) of a hemisphere $= 2\pi r^2$
• Area of the circular base of the hemisphere $= \pi r^2$

Putting them together:

$\text{Surface area of solid} = \underbrace{6a^2}_{\text{cube}} + \underbrace{2\pi r^2}_{\text{CSA of hemisphere}} - \underbrace{\pi r^2}_{\text{base circle covered}}$

The two hemisphere terms combine, since $2\pi r^2 - \pi r^2 = \pi r^2$:

$= 6a^2 + \pi r^2$

Now we substitute the values, with side of cube $a = \mathbf{7 \text{ cm}}$ and radius $r = \mathbf{3.5 \text{ cm}}$, using $\pi = \dfrac{22}{7}$:

$= 6 \times (7)^2 + \frac{22}{7} \times (3.5)^2$

$= 6 \times 49 + \frac{22}{7} \times 12.25$

Here $12.25 \div 7 = 1.75$, so:

$= 294 + 22 \times 1.75$

$= 294 + 38.5$

$\boxed{332.5 \text{ cm}^2}$

Answer

(i) The greatest diameter the hemisphere can have is $\mathbf{7 \text{ cm}}$. (ii) The surface area of the solid is $\mathbf{332.5 \text{ cm}^2}$.