The height of a right prism is 15 cm. Its base is a triangle with sides measuring 10 cm, 17 cm and 9 cm. The volume of the prism is

Options

A. 360 `cm^3`
B. 540 `cm^3`
C. 540 `m^3`
D. None of these

Answer

To find the volume of the right prism with a triangular base, we will follow these steps:

### Step 1: Identify the given values
- Height of the prism (h) = 15 cm
- Sides of the triangular base: a = 10 cm, b = 17 cm, c = 9 cm

### Step 2: Calculate the semi-perimeter (s) of the triangle
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{10 + 17 + 9}{2} = \frac{36}{2} = 18 \text{ cm} \]

### Step 3: Use Heron's formula to calculate the area of the triangle
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values:
\[ A = \sqrt{18 \times (18 - 10) \times (18 - 17) \times (18 - 9)} \]
Calculating each term:
- \( s - a = 18 - 10 = 8 \)
- \( s - b = 18 - 17 = 1 \)
- \( s - c = 18 - 9 = 9 \)

Now substituting back into the area formula:
\[ A = \sqrt{18 \times 8 \times 1 \times 9} \]
Calculating the product:
\[ A = \sqrt{18 \times 72} = \sqrt{1296} = 36 \text{ cm}^2 \]

### Step 4: Calculate the volume of the prism
The volume \( V \) of the prism is given by the formula:
\[ V = \text{Base Area} \times \text{Height} \]
Substituting the values:
\[ V = 36 \text{ cm}^2 \times 15 \text{ cm} = 540 \text{ cm}^3 \]

### Final Answer
The volume of the prism is \( 540 \text{ cm}^3 \).

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