Question Analysis:
We are given the dimensions of a column and we need to find its slenderness ratio. The slenderness ratio is a measure of how slender a column is and is calculated by dividing the length of the column by the radius of gyration.

Given:
- Cross-sectional dimensions: 10 mm x 20 mm
- Length of the column: 1 m

Find:
Slenderness ratio of the column

Solution:

Step 1: Calculate the Moment of Inertia (I)
The moment of inertia is a property of the cross-sectional shape and is given by the formula:

I = (b * h^3) / 12

Where:
- b is the width of the cross-section (10 mm)
- h is the height of the cross-section (20 mm)

Substituting the values, we get:

I = (10 * 20^3) / 12
I = 6666.67 mm^4

Step 2: Calculate the Radius of Gyration (r)
The radius of gyration is a measure of how the area of the cross-section is distributed from the centroid and is calculated by taking the square root of the ratio of the moment of inertia to the cross-sectional area.

The cross-sectional area (A) is given by:

A = b * h

Substituting the values, we get:

A = 10 mm * 20 mm
A = 200 mm^2

The radius of gyration (r) is then calculated as:

r = sqrt(I / A)
r = sqrt(6666.67 mm^4 / 200 mm^2)
r = sqrt(33.33335 mm^2)
r = 5.7735 mm

Step 3: Calculate the Slenderness Ratio
The slenderness ratio (λ) is calculated by dividing the length of the column (L) by the radius of gyration (r).

λ = L / r
λ = 1000 mm / 5.7735 mm
λ ≈ 173.2

The slenderness ratio of the column is approximately 173.2. However, none of the given options match this value exactly.

Step 4: Determine the Closest Option
To determine the closest option, we can compare the calculated slenderness ratio to each of the given options.

- Option (a): 200 - The calculated slenderness ratio is less than 200.
- Option (b): 346 - The calculated slenderness ratio is closer to 346.
- Option (c): 477 - The calculated slenderness ratio is greater than 477.
- Option (d): 1000 - The calculated slenderness ratio is greater than 1000.

Therefore, the closest option to the calculated slenderness ratio is option (b) - 346.