A 2-D Surface profile is obtained over a sampling length by using a contact type stylus profilometer as shown in figure. A line AA parallel to the general lay of the trace is considered. The true heights from AA to the peaks & valleys (in µm) in the trace are also shown in figure. (Not to scale). The ten-point height average in µm is

The correct option is 4.57.

Concept and Formula:
The ten-point height average ($R_z$) of a surface profile is defined as the average difference between the five highest peaks and the five deepest valleys within the sampling length, measured from a reference line (in this case, the line AA parallel to the general lay of the trace).
The formula is given by:

$R_z=\frac{\sum _{i=1}^5{y}_{pi}-\sum _{i=1}^5{y}_{vi}}{5}$

where:
• $y_{pi}$ represents the heights of the 5 highest peaks.
• $y_{vi}$ represents the heights of the 5 deepest (lowest) valleys measured from line AA.

Step 1: Identify and sort all peaks and valleys from the profile image
By analyzing the surface profile shown in the image, we can extract the heights of all the peaks and valleys relative to the reference line AA:
• All Peak Heights (in µm): 5.35, 6.7, 5.28, 4.32, 8.3, 6.3, 4.96, 6.9, 6.2, 5.54
• All Valley Heights (in µm): 2.81, 1.92, 3.71, 2.82, 4.2, 1.97, 2.01, 2.9, 4.26, 4.3

Step 2: Select the 5 highest peaks
Sorting the peak heights in descending order, we get:
8.3, 6.9, 6.7, 6.3, 6.2, 5.54, 5.35, 5.28, 4.96, 4.32
The five highest peaks are:
• $y_{p1}=8.3$ µm
• $y_{p2}=6.9$ µm
• $y_{p3}=6.7$ µm
• $y_{p4}=6.3$ µm
• $y_{p5}=6.2$ µm

Sum of the 5 highest peaks:

$\sum y_p=8.3+6.9+6.7+6.3+6.2=34.4\mathrm{\mu m}$

Step 3: Select the 5 deepest valleys
The deepest valleys correspond to the lowest values (closest distance to the reference line AA). Sorting the valley heights in ascending order, we get:
1.92, 1.97, 2.01, 2.81, 2.82, 2.9, 3.71, 4.2, 4.26, 4.3
The five deepest valleys are:
• $y_{v1}=1.92$ µm
• $y_{v2}=1.97$ µm
• $y_{v3}=2.01$ µm
• $y_{v4}=2.81$ µm
• $y_{v5}=2.82$ µm

Sum of the 5 deepest valleys:

$\sum y_v=1.92+1.97+2.01+2.81+2.82=11.53\mathrm{\mu m}$

Step 4: Calculate the ten-point height average ($R_z$)
Substitute the summed values into the formula:

$R_z=\frac{34.4-11.53}{5}$

$R_z=\frac{22.87}{5}=4.574\mathrm{\mu m}$

Rounding to two decimal places, we find the ten-point average roughness value is 4.57 µm.