If two adjacent sides of a rectangular parallelopiped are 1 cm and 2 cm and the total surface area of the parallelopiped is 22 square cm, then the diagonal of the parallelopied is

Options

A. `sqrt(10) cm`
B. `2sqrt(3) cm`
C. `sqrt(14) cm`
D. 4 cm

Answer

To find the diagonal of the rectangular parallelepiped, we will follow these steps:

### Step 1: Understand the dimensions
Let the dimensions of the rectangular parallelepiped be:
- Length (l) = 1 cm
- Width (w) = 2 cm
- Height (h) = ?

### Step 2: Use the formula for total surface area
The formula for the total surface area (SA) of a rectangular parallelepiped is given by:
\[ SA = 2(lw + lh + wh) \]
Given that the total surface area is 22 cm², we can set up the equation:
\[ 2(1 \cdot 2 + 1 \cdot h + 2 \cdot h) = 22 \]

### Step 3: Simplify the equation
Now, simplify the equation:
\[ 2(2 + h + 2h) = 22 \]
\[ 2(2 + 3h) = 22 \]
Dividing both sides by 2:
\[ 2 + 3h = 11 \]

### Step 4: Solve for height (h)
Now, isolate h:
\[ 3h = 11 - 2 \]
\[ 3h = 9 \]
\[ h = 3 \text{ cm} \]

### Step 5: Calculate the diagonal
The formula for the diagonal (d) of a rectangular parallelepiped is:
\[ d = \sqrt{l^2 + w^2 + h^2} \]
Substituting the values we have:
\[ d = \sqrt{1^2 + 2^2 + 3^2} \]
\[ d = \sqrt{1 + 4 + 9} \]
\[ d = \sqrt{14} \]

### Final Answer
Thus, the diagonal of the rectangular parallelepiped is:
\[ d = \sqrt{14} \text{ cm} \]

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