The perimeter of the triangular base of a right prism is 60 cm and the sides of the base are in the ratio 5:12: 13. Then, its volume will be (height of the prism being 50 cm)

Options

A. `6000 cm^(3) `
B. `6600 cm^(3) `
C. `5400 cm^(3) `
D. `9600 cm^(3) `

Answer

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

### Step 1: Determine the sides of the triangle
The sides of the triangular base are in the ratio 5:12:13. Let's denote the sides as:
- Side 1 = 5x
- Side 2 = 12x
- Side 3 = 13x

### Step 2: Calculate the perimeter of the triangle
The perimeter of the triangle is given as 60 cm. Therefore, we can set up the equation:
\[ 5x + 12x + 13x = 60 \]
\[ 30x = 60 \]

### Step 3: Solve for x
Now, we solve for x:
\[ x = \frac{60}{30} = 2 \]

### Step 4: Find the lengths of the sides
Now we can find the actual lengths of the sides:
- Side 1 = \( 5x = 5 \times 2 = 10 \) cm
- Side 2 = \( 12x = 12 \times 2 = 24 \) cm
- Side 3 = \( 13x = 13 \times 2 = 26 \) cm

### Step 5: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) of the triangle is:
\[ s = \frac{Perimeter}{2} = \frac{60}{2} = 30 \text{ cm} \]

### Step 6: Use Heron's formula to find the area of the triangle
The area \( A \) of the triangle can be calculated using Heron's formula:
\[ A = \sqrt{s(s - a)(s - b)(s - c)} \]
Where \( a, b, c \) are the sides of the triangle.
Substituting the values:
\[ A = \sqrt{30(30 - 10)(30 - 24)(30 - 26)} \]
\[ A = \sqrt{30 \times 20 \times 6 \times 4} \]
\[ A = \sqrt{30 \times 480} \]
\[ A = \sqrt{14400} = 120 \text{ cm}^2 \]

### Step 7: Calculate the volume of the prism
The volume \( V \) of the prism is given by the formula:
\[ V = \text{Area of base} \times \text{Height} \]
Substituting the area we found and the height (50 cm):
\[ V = 120 \times 50 = 6000 \text{ cm}^3 \]

### Final Answer
The volume of the prism is \( 6000 \text{ cm}^3 \).

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