Let ABCDEF be a prism whose base is a right angled triangle where sides adjacent to `90^(@)` are 9 cm and 12 cm. If the cost of painting the prism is Rs. 151.20, at the rate of 20 paise per sq cm then the height of the prism is :

Options

A. 17 cm
B. 18 cm
C. 15 cm
D. 16 cm

Answer

To find the height of the prism, we can follow these steps:

### Step 1: Calculate the Area of the Prism
Given the total cost of painting the prism is Rs. 151.20 and the rate is 20 paise (or Rs. 0.20) per square centimeter, we can find the area of the surface of the prism using the formula:

\[ \text{Area} = \frac{\text{Cost}}{\text{Rate}} = \frac{151.20}{0.20} = 756 \text{ cm}^2 \]

### Step 2: Find the Perimeter of the Base Triangle
The base of the prism is a right-angled triangle with sides 9 cm and 12 cm. We need to find the hypotenuse using the Pythagorean theorem:

\[ \text{Hypotenuse}^2 = 9^2 + 12^2 = 81 + 144 = 225 \implies \text{Hypotenuse} = \sqrt{225} = 15 \text{ cm} \]

Now, we can find the perimeter of the triangle:

\[ \text{Perimeter} = 9 + 12 + 15 = 36 \text{ cm} \]

### Step 3: Calculate the Area of the Base Triangle
The area of a right-angled triangle can be calculated using the formula:

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 9 \times 12 = 54 \text{ cm}^2 \]

### Step 4: Use the Surface Area Formula for the Prism
The surface area \( A \) of the prism can be expressed as:

\[ A = \text{Perimeter of base} \times \text{Height} + 2 \times \text{Area of base} \]

Substituting the known values:

\[ 756 = 36h + 2 \times 54 \]

Calculating \( 2 \times 54 \):

\[ 2 \times 54 = 108 \]

So the equation becomes:

\[ 756 = 36h + 108 \]

### Step 5: Solve for Height \( h \)
Rearranging the equation to isolate \( h \):

\[ 756 - 108 = 36h \]
\[ 648 = 36h \]
\[ h = \frac{648}{36} = 18 \text{ cm} \]

### Conclusion
The height of the prism is **18 cm**.

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