Finding the Total Surface Area of a Solid Hemisphere from its Volume

The problem asks us to find the total surface area of a solid hemisphere given its volume. We are provided with the volume and asked to use the value of \(\pi\) as \(\frac{22}{7}\).

Given Information

• Volume of the solid hemisphere = 19,404 cm³
• Value of \(\pi\) to use = \(\frac{22}{7}\)

Finding the Radius of the Hemisphere

The formula for the volume of a solid hemisphere is given by:

\(V = \frac{2}{3} \pi r^3\)

Where \(V\) is the volume and \(r\) is the radius of the hemisphere.

We are given \(V = 19404 \text{ cm}^3\). Substituting the given values into the formula:

\(19404 = \frac{2}{3} \times \frac{22}{7} \times r^3\)

\(19404 = \frac{44}{21} r^3\)

To find \(r^3\), we can rearrange the equation:

\(r^3 = 19404 \times \frac{21}{44}\)

Let's perform the division of 19404 by 44:

\(19404 \div 44\)

We can divide both by 4 first: \(19404 \div 4 = 4851\), and \(44 \div 4 = 11\). So we have \(4851 \div 11\).

\(4851 \div 11 = 441\)

So, the equation becomes:

\(r^3 = 441 \times 21\)

We know that \(441 = 21^2\). So,

\(r^3 = 21^2 \times 21\)

\(r^3 = 21^3\)

Taking the cube root of both sides:

\(r = 21 \text{ cm}\)

The radius of the solid hemisphere is 21 cm.

Calculating the Total Surface Area

The total surface area of a solid hemisphere includes both the curved surface area and the area of the circular base. The formula for the total surface area (TSA) is:

\(TSA = 3 \pi r^2\)

Where \(r\) is the radius we just found (21 cm) and \(\pi = \frac{22}{7}\).

Substitute the values into the formula:

\(TSA = 3 \times \frac{22}{7} \times (21)^2\)

\(TSA = 3 \times \frac{22}{7} \times 441\)

Divide 441 by 7:

\(441 \div 7 = 63\)

So, the calculation becomes:

\(TSA = 3 \times 22 \times 63\)

\(TSA = 66 \times 63\)

Now, multiply 66 by 63:

\(66 \times 63 = 4158\)

The total surface area of the solid hemisphere is 4158 cm².

Summary of Steps

1. Use the given volume of the solid hemisphere and the formula \(V = \frac{2}{3} \pi r^3\) to find the radius \(r\).
2. Substitute the calculated radius \(r\) into the formula for the total surface area of a solid hemisphere, \(TSA = 3 \pi r^2\).
3. Calculate the final value of the total surface area.

Revision Table: Hemisphere Formulas

Additional Information on Hemispheres and Surface Area

A hemisphere is exactly half of a sphere. When we talk about the surface area of a hemisphere, it's important to distinguish between the curved surface area and the total surface area, especially if it's a solid object.

• Curved Surface Area: This is the area of the rounded part only. For a hemisphere, this is half the surface area of the full sphere, which is \(2 \pi r^2\).
• Total Surface Area: For a *solid* hemisphere, this includes the curved surface area plus the area of the flat circular base. The base is a circle with radius \(r\), so its area is \(\pi r^2\). Thus, the total surface area of a solid hemisphere is \(2 \pi r^2 + \pi r^2 = 3 \pi r^2\). If the hemisphere is hollow (like a bowl without a lid), its total surface area might just be the curved surface area, or it could include the area of a thin rim, depending on the specific definition in a problem. In this question, it specifies a "solid hemisphere", so we must include the base area.
• Volume: The volume of a hemisphere is always half the volume of the corresponding sphere. The volume of a sphere is \(\frac{4}{3} \pi r^3\), so the volume of a hemisphere is \(\frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3\). This formula applies whether the hemisphere is solid or hollow (as long as we're calculating the space it occupies).

Understanding these distinct formulas is crucial for solving problems involving hemispheres in geometry and mensuration.