The base of a right prism is a trapezium. The length of the parallel sides are 8 cm and 14 cm and the distance between the parallel sides is 8 cm. If the volume of the prism is `1056 cm^(3)`, then the height of the prism is

Options

A. 44 cm
B. 16.5 cm
C. 12 cm
D. 10.56 cm

Answer

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

### Step 1: Calculate the area of the trapezium base.
The formula for the area \( A \) of a trapezium is given by:

\[ A = \frac{1}{2} \times (a + b) \times h \]

where \( a \) and \( b \) are the lengths of the parallel sides, and \( h \) is the distance between them.

Given:
- \( a = 8 \, \text{cm} \)
- \( b = 14 \, \text{cm} \)
- \( h = 8 \, \text{cm} \)

Substituting the values into the formula:

\[ A = \frac{1}{2} \times (8 + 14) \times 8 \]

Calculating the sum of the parallel sides:

\[ A = \frac{1}{2} \times 22 \times 8 \]

Calculating the area:

\[ A = \frac{1}{2} \times 176 = 88 \, \text{cm}^2 \]

### Step 2: Use the volume formula of the prism.
The volume \( V \) of a prism is given by:

\[ V = \text{Base Area} \times \text{Height} \]

Given the volume of the prism:

\[ V = 1056 \, \text{cm}^3 \]

Substituting the area of the trapezium and height \( H \):

\[ 1056 = 88 \times H \]

### Step 3: Solve for the height \( H \).
To find \( H \), we rearrange the equation:

\[ H = \frac{1056}{88} \]

Calculating \( H \):

\[ H = 12 \, \text{cm} \]

### Conclusion:
The height of the prism is \( 12 \, \text{cm} \).

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