To obtain undisturbed clay soil sample, an Area Ratio of 10 % needs to be achieved for a thin walled sampling tube. If the outer diameter of the tube is 50.8 mm, the inner diameter (in mm) is _________ (rounded off to one decimal place).

The correct option is 48.4.

Understanding Area Ratio:
In soil exploration, soil samples are obtained using sampling tubes. To get an "undisturbed" sample, the disturbance caused by the thickness of the cutting edge of the sampler needs to be minimized. The area ratio is a key parameter that measures this thickness, representing the ratio of the volume of soil displaced by the sampler tube to the volume of the sample. To obtain undisturbed samples of clay, a thin-walled sampler with a low area ratio (typically 10% or less) is required.

The formula for the area ratio (denoted as $A_r$) is given by:

$$A_r=\frac{D_o^2-D_i^2}{D_i^2}\times 100$$

Where:
• $D_o$ is the outer diameter of the sampling tube.
• $D_i$ is the inner diameter of the sampling tube.

Given Data:
• Area Ratio, $A_r=10\%$
• Outer diameter, $D_o=50.8\text{mm}$

Step-by-Step Calculation:
Substituting the given values into the area ratio formula, we express the percentage as a fraction:

$$0.10=\frac{{\left(50.8\right)}^2-D_i^2}{D_i^2}$$

Next, we multiply both sides by $D_i^2$ to eliminate the fraction:

$$0.10D_i^2={\left(50.8\right)}^2-D_i^2$$

Rearranging the equation to group the $D_i^2$ terms on one side:

$$0.10D_i^2+D_i^2={\left(50.8\right)}^2$$

$$1.10D_i^2=2580.64$$

Solving for $D_i^2$:

$$D_i^2=\frac{2580.64}{1.10}$$

$$D_i^2=2346.036$$

Taking the square root of both sides to find $D_i$:

$$D_i=\sqrt{2346.036}\approx 48.436\text{mm}$$

Rounding to one decimal place gives:
$D_i=48.4\text{mm}$