The base of a right prism is an equilateral triangle of area `173 cm^(2)` and the volume of the prism is `10380 cm^(3)`. The area of the lateral surface of the prism is (use `sqrt(3) = 1.73`)

Options

A. `1200 cm^(2)`
B. `2400 cm^(2)`
C. `3600 cm^(2)`
D. `4380 cm^(2)`

Answer

To find the lateral surface area of the right prism with an equilateral triangle as its base, we will follow these steps:

### Step 1: Calculate the side length of the equilateral triangle
The area \( A \) of an equilateral triangle can be expressed as:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
where \( a \) is the side length of the triangle. Given that the area is \( 173 \, \text{cm}^2 \), we can set up the equation:
\[ \frac{\sqrt{3}}{4} a^2 = 173 \]
Substituting \( \sqrt{3} = 1.73 \):
\[ \frac{1.73}{4} a^2 = 173 \]
Multiplying both sides by \( 4 \):
\[ 1.73 a^2 = 692 \]
Now, divide both sides by \( 1.73 \):
\[ a^2 = \frac{692}{1.73} \approx 400 \]
Taking the square root:
\[ a = \sqrt{400} = 20 \, \text{cm} \]

### Step 2: Calculate the height of the prism
The volume \( V \) of the prism is given by:
\[ V = \text{Area of base} \times \text{Height} \]
We know the volume is \( 10,380 \, \text{cm}^3 \) and the area of the base is \( 173 \, \text{cm}^2 \):
\[ 10,380 = 173 \times h \]
Solving for \( h \):
\[ h = \frac{10,380}{173} \approx 60 \, \text{cm} \]

### Step 3: Calculate the perimeter of the base
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3a \]
Substituting \( a = 20 \, \text{cm} \):
\[ P = 3 \times 20 = 60 \, \text{cm} \]

### Step 4: Calculate the lateral surface area
The lateral surface area \( L \) of the prism is given by:
\[ L = \text{Perimeter of base} \times \text{Height} \]
Substituting the values we found:
\[ L = 60 \times 60 = 3600 \, \text{cm}^2 \]

### Final Answer
The area of the lateral surface of the prism is \( 3600 \, \text{cm}^2 \).

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