A right prism stands on a base 6 cm equilateral triangle and its volume is `81sqrt(3) cm^(3)`. The height (in cm) of the prism is

Options

A. 9
B. 10
C. 12
D. 15

Answer

To find the height of the right prism with a base that is an equilateral triangle, we can follow these steps:

### Step 1: Understand the formula for the volume of a prism
The volume \( V \) of a prism is given by the formula:
\[ V = \text{Area of base} \times \text{Height} \]

### Step 2: Identify the given values
From the problem, we know:
- Volume \( V = 81\sqrt{3} \, \text{cm}^3 \)
- The base of the prism is an equilateral triangle with side length \( s = 6 \, \text{cm} \).

### Step 3: Calculate the area of the base (equilateral triangle)
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
Substituting the side length \( s = 6 \, \text{cm} \):
\[ A = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \, \text{cm}^2 \]

### Step 4: Set up the equation for the volume
Now, we can substitute the area of the base into the volume formula:
\[ 81\sqrt{3} = 9\sqrt{3} \times h \]

### Step 5: Solve for the height \( h \)
To find \( h \), we can divide both sides of the equation by \( 9\sqrt{3} \):
\[ h = \frac{81\sqrt{3}}{9\sqrt{3}} = \frac{81}{9} = 9 \, \text{cm} \]

### Conclusion
The height of the prism is \( 9 \, \text{cm} \).

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