Base of a right prism is a rectangle, the ratio of whose lengh and breadth is 3:2. If the height of the prism is 12 cm and total surface area is 288 sq. cm. the volume of the prism is :

Options

A. a) `291 cm^(3)`
B. b) `288 cm^(3)`
C. c) `290 cm^(3)`
D. d) `286 cm^(3)`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the given information
We know that the base of the right prism is a rectangle, with the ratio of length (L) to breadth (B) as 3:2. The height (H) of the prism is given as 12 cm, and the total surface area (TSA) is 288 sq. cm.

### Step 2: Express length and breadth in terms of a variable
Let’s denote the length and breadth using a variable \( x \):
- Length \( L = 3x \)
- Breadth \( B = 2x \)

### Step 3: Write the formula for the total surface area of the prism
The formula for the total surface area (TSA) of a right prism is given by:
\[ \text{TSA} = 2(LB + BH + HL) \]
Substituting the values of \( L \), \( B \), and \( H \):
\[ \text{TSA} = 2((3x)(2x) + (2x)(12) + (12)(3x)) \]

### Step 4: Simplify the equation
Calculating each term:
- \( LB = (3x)(2x) = 6x^2 \)
- \( BH = (2x)(12) = 24x \)
- \( HL = (12)(3x) = 36x \)

Now substituting back into the TSA formula:
\[ \text{TSA} = 2(6x^2 + 24x + 36x) = 2(6x^2 + 60x) \]
\[ \text{TSA} = 12x^2 + 120x \]

### Step 5: Set the TSA equal to the given value
We know the TSA is 288 sq. cm:
\[ 12x^2 + 120x = 288 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 12x^2 + 120x - 288 = 0 \]

### Step 7: Simplify the equation
Dividing the entire equation by 12:
\[ x^2 + 10x - 24 = 0 \]

### Step 8: Factor the quadratic equation
To factor the equation:
\[ (x + 12)(x - 2) = 0 \]
This gives us two possible solutions for \( x \):
- \( x = -12 \) (not valid since dimensions cannot be negative)
- \( x = 2 \)

### Step 9: Find the dimensions of the base
Now substituting \( x = 2 \):
- Length \( L = 3x = 3(2) = 6 \) cm
- Breadth \( B = 2x = 2(2) = 4 \) cm

### Step 10: Calculate the volume of the prism
The volume \( V \) of the prism is given by:
\[ V = L \times B \times H \]
Substituting the values:
\[ V = 6 \times 4 \times 12 \]
Calculating this gives:
\[ V = 288 \text{ cm}^3 \]

### Final Answer
The volume of the prism is **288 cm³**.

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