Microchip etching roughly follows the relation 𝒅 = 𝟏𝟔. 𝟐 −𝟏𝟔. 𝟐 𝐞𝐱𝐩(−𝟎. 𝟎𝟐𝟏𝒕) where d is the depth of etch in micrometers (μm) and t is the time of etch in seconds (s). The said relation is only valid for t < 200s. • What are the units associated with the numbers 16.2 and 0.021? • Convert the said relation such that d becomes expressed in inches and t can be used in minutes. • Solve for depth of etch (in) if the time of etching was found out to be 4 minutes.

To analyze the given etching relation:

\[
d = 16.2 - 16.2 \cdot \exp(-0.021t)
\]
where \( d \) is the depth of etch in micrometers (μm) and \( t \) is the time in seconds (s), we need to address the following points.

### 1. Units of 16.2 and 0.021:
- **16.2:** The first term, \( 16.2 \), represents a depth of etch in **micrometers (μm)**. This is because \( d \) is in micrometers, and so the constant must have the same unit.
- **0.021:** The second term, \( 0.021 \), is associated with the exponential term and must be in **seconds\(^{-1}\)**, as \( t \) is given in seconds and the exponent in the exponential function must be dimensionless.

Thus:
- 16.2 has units of micrometers (μm),
- 0.021 has units of seconds\(^{-1}\).

---

### 2. Converting to inches and minutes:

To convert the formula such that \( d \) is in inches and \( t \) is in minutes, we follow these steps:

- **Converting depth (d) from micrometers to inches:**
  
  1 inch = 25,400 micrometers, so to convert micrometers to inches, we divide by 25,400.
  
  Thus, \( d_{\text{inches}} = \frac{d_{\text{μm}}}{25,400} \).

- **Converting time (t) from seconds to minutes:**

  There are 60 seconds in a minute, so to convert time from seconds to minutes, we replace \( t_{\text{seconds}} \) by \( t_{\text{minutes}} \times 60 \).

### New relation:
Given these conversions, the original relation becomes:

\[
d_{\text{inches}} = \frac{16.2 - 16.2 \cdot \exp(-0.021 \cdot (t_{\text{minutes}} \cdot 60))}{25,400}
\]

Simplifying the exponent:

\[
d_{\text{inches}} = \frac{16.2 - 16.2 \cdot \exp(-1.26 t_{\text{minutes}})}{25,400}
\]

This is the equation for \( d \) in inches and \( t \) in minutes.

---

### 3. Solve for depth when \( t = 4 \) minutes:

Using the new equation and substituting \( t_{\text{minutes}} = 4 \):

\[
d_{\text{inches}} = \frac{16.2 - 16.2 \cdot \exp(-1.26 \cdot 4)}{25,400}
\]

Now let's calculate the depth in inches.

The depth of the etch after 4 minutes is approximately \( 0.000634 \) inches.

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Would you like further details on any part of the process, or do you have additional questions?

Here are five related questions to consider:
1. How does the etching depth change if the time increases?
2. What would the depth be in micrometers after 4 minutes instead of inches?
3. How does the exponential term influence the rate of etching over time?
4. How would the depth change if the time were halved to 2 minutes?
5. What happens to the etching depth as time approaches 200 seconds?

**Tip:** Exponential functions like the one used here often represent processes where growth or decay slows over time.

Explore how to convert the microchip etching equation from micrometers and seconds to inches and minutes. Learn step-by-step how the depth of etch changes over time, with a practical example where etching takes place for 4 minutes. Suitable for advanced physics or engineering students.

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