Calculating Hemisphere Radius from Surface Area and Volume

This question asks us to find the radius of a hemisphere given its curved surface area and its volume. We can use the standard formulas for the curved surface area and volume of a hemisphere to solve this problem.

The given values are:

• Curved Surface Area (CSA) of hemisphere = 2772 cm²
• Volume (V) of hemisphere = 19404 cm³

The formulas for a hemisphere with radius 'r' are:

• Curved Surface Area (CSA) $= 2\pi r^2$
• Volume (V) $= \frac{2}{3}\pi r^3$

We can find the radius using either of the given pieces of information. Let's use the Curved Surface Area formula first.

Finding Radius Using Curved Surface Area

We are given that the Curved Surface Area is 2772 cm². Using the formula:

CSA = 2πr²

Substitute the given value:

2772 = 2πr²

To find r², rearrange the equation:

r² = \(\frac{2772}{2π}\)

r² = \(\frac{1386}{\pi}\)

Using the common approximation for π as \(\frac{22}{7}\):

r² = \(\frac{1386}{\frac{22}{7}}\)

r² = \(\frac{1386 \times 7}{22}\)

Now, perform the division and multiplication:

1386 ÷ 22 = 63

r² = 63 \(\times\) 7

r² = 441

To find the radius 'r', take the square root of both sides:

r = √441

r = 21

So, the radius of the hemisphere is 21 cm.

Verifying Radius Using Volume

Let's verify this result using the Volume formula. The given volume is 19404 cm³. Using the formula:

V = \(\frac{2}{3}πr³\)

Substitute the given value:

19404 = \(\frac{2}{3}πr³\)

To find r³, rearrange the equation:

r³ = \(\frac{19404 \times 3}{2\pi}\)

r³ = \(\frac{9702 \times 3}{\pi}\)

Using π ≈ \(\frac{22}{7}\):

r³ =\(\frac{9702 \times 3}{\frac{22}{7}}\)

r³ = \(\frac{9702 \times 3 \times 7}{22}\)

Perform the calculations:

9702 ÷ 22 = 441

r³ = 441 \(\times\) 3 \(\times\) 7

r³ = 441 \(\times\) 21

r³ = 9261

To find the radius 'r', take the cube root of both sides:

r = \(\sqrt[3]{9261}\)

We know that 21 × 21 × 21 = 441 × 21 = 9261.

r = 21

Both methods yield the same radius, 21 cm.

Final Hemisphere Radius

Based on the given curved surface area and volume, the radius of the hemisphere is 21 cm.

Revision Table: Hemisphere Formulas

Understanding the key formulas is crucial for solving problems related to hemispheres.

Additional Information on Hemispheres

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.

• Curved Surface Area: This is the area of the curved part, which is half the surface area of the full sphere (4πr² / 2 = 2πr²).
• Total Surface Area: This includes the curved surface area plus the area of the flat circular base. The base area is πr² (area of a circle). So, the total surface area is 2πr² + πr² = 3πr².
• Volume: The volume of a hemisphere is half the volume of a full sphere (\(\frac{4}{3}πr³ / 2 = \frac{2}{3}πr³\)).

These formulas are fundamental in mensuration problems involving spheres and hemispheres. It's often helpful to remember the formulas for a full sphere first and then derive the hemisphere formulas by dividing by two (for volume and curved surface area) and adding the base area (for total surface area).