The curved surface area of a hemisphere is 27.72 square cm and volume is 19.404 cubic cm, find its radius.

Options

A. 4.2 cm
B. 2.1 cm
C. 0.7 cm
D. 1.05 cm

Answer

To find the radius of the hemisphere given its curved surface area and volume, we can use the formulas for both.

### Step 1: Use the formula for the curved surface area of a hemisphere
The formula for the curved surface area (CSA) of a hemisphere is:
\[ \text{CSA} = 2\pi r^2 \]
Given that the curved surface area is 27.72 cm², we can set up the equation:
\[ 2\pi r^2 = 27.72 \]

### Step 2: Solve for \( r^2 \)
First, we can isolate \( r^2 \):
\[ r^2 = \frac{27.72}{2\pi} \]
Now, substituting the value of \( \pi \) (approximately 3.14):
\[ r^2 = \frac{27.72}{2 \times 3.14} = \frac{27.72}{6.28} \approx 4.41 \]

### Step 3: Calculate \( r \)
Now, to find \( r \), we take the square root of \( r^2 \):
\[ r = \sqrt{4.41} \approx 2.1 \text{ cm} \]

### Step 4: Verify with the volume formula
To ensure our value for \( r \) is correct, we can verify it using the volume formula for a hemisphere:
\[ \text{Volume} = \frac{2}{3}\pi r^3 \]
Substituting \( r \approx 2.1 \):
\[ \text{Volume} = \frac{2}{3} \pi (2.1)^3 \]
Calculating \( (2.1)^3 \):
\[ (2.1)^3 = 9.261 \]
Now substituting back:
\[ \text{Volume} \approx \frac{2}{3} \times 3.14 \times 9.261 \approx 19.404 \text{ cm}^3 \]
This confirms our radius is correct.

### Final Answer
The radius of the hemisphere is approximately \( 2.1 \text{ cm} \).

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