The total surface are of a hemisphere is very nearly equal to that of an equilateral triangle. The side of the triangle is how many times (approximately) of the radius of the hemisphere?

Options

A. `((8pi)/sqrt3)0.5`
B. `((4pi)/3)0.5`
C. `(2pi sqrt3)0.5`
D. `(4pisqrt3)0.5`

Answer

To solve the problem, we need to find the relationship between the side of an equilateral triangle and the radius of a hemisphere, given that their total surface areas are equal.

### Step-by-Step Solution:

1. **Understand the Total Surface Area of a Hemisphere**:
The total surface area (TSA) of a hemisphere is given by the formula:
\[ \text{TSA of Hemisphere} = 3\pi r^2 \]
where \( r \) is the radius of the hemisphere.

2. **Understand the Total Surface Area of an Equilateral Triangle**:
The total surface area of an equilateral triangle can be represented as:
\[ \text{Area of Equilateral Triangle} = \frac{\sqrt{3}}{4} a^2 \]
where \( a \) is the length of one side of the triangle.

3. **Set the Two Areas Equal**:
According to the problem, the total surface area of the hemisphere is approximately equal to the area of the equilateral triangle:
\[ 3\pi r^2 = \frac{\sqrt{3}}{4} a^2 \]

4. **Rearranging the Equation**:
To find \( a \) in terms of \( r \), we can rearrange the equation:
\[ a^2 = \frac{4 \cdot 3\pi r^2}{\sqrt{3}} \]
Simplifying this gives:
\[ a^2 = \frac{12\pi r^2}{\sqrt{3}} \]

5. **Taking the Square Root**:
Now, take the square root of both sides to find \( a \):
\[ a = \sqrt{\frac{12\pi}{\sqrt{3}}} \cdot r \]

6. **Simplifying the Expression**:
We can simplify \( \sqrt{\frac{12\pi}{\sqrt{3}}} \):
\[ a = \sqrt{12\pi} \cdot \frac{1}{\sqrt[4]{3}} \cdot r \]
This can be further simplified to:
\[ a \approx 2\sqrt{3\pi} \cdot r \]

7. **Calculating the Approximate Value**:
To find the approximate numerical value, we can substitute \( \pi \approx 3.14 \):
\[ a \approx 2\sqrt{3 \cdot 3.14} \cdot r \approx 2\sqrt{9.42} \cdot r \approx 2 \cdot 3.07 \cdot r \approx 6.14r \]

Thus, the side of the triangle is approximately \( 6.14 \) times the radius of the hemisphere.

### Final Answer:
The side of the triangle is approximately \( 6.14 \) times the radius of the hemisphere.