0.1093 + (16° 43' 38.44"- 16° 33' 43") x ((0.1129-0.1093 )/(16° 55' 41"-16° 33' 43"))

Let's break down the given expression and solve it step by step.

### Expression:
\[
0.1093 + \left(16^\circ 43' 38.44" - 16^\circ 33' 43"\right) \times \frac{0.1129 - 0.1093}{16^\circ 55' 41" - 16^\circ 33' 43"}
\]

### Step 1: Convert the angles to decimal degrees.
1 degree \((^\circ)\) = 60 minutes \((')\) = 3600 seconds \((\")\).

#### a) \( 16^\circ 43' 38.44" \) to decimal degrees:
\[
16^\circ + \frac{43'}{60} + \frac{38.44"}{3600} = 16 + 0.7167 + 0.0107 = 16.7274^\circ
\]

#### b) \( 16^\circ 33' 43" \) to decimal degrees:
\[
16^\circ + \frac{33'}{60} + \frac{43"}{3600} = 16 + 0.55 + 0.0119 = 16.5619^\circ
\]

#### c) \( 16^\circ 55' 41" \) to decimal degrees:
\[
16^\circ + \frac{55'}{60} + \frac{41"}{3600} = 16 + 0.9167 + 0.0114 = 16.9281^\circ
\]

### Step 2: Substitute the decimal degrees back into the expression.
Now that we have the angles in decimal degrees, the expression becomes:

\[
0.1093 + (16.7274 - 16.5619) \times \frac{0.1129 - 0.1093}{16.9281 - 16.5619}
\]

### Step 3: Simplify the differences.
\[
16.7274 - 16.5619 = 0.1655
\]
\[
16.9281 - 16.5619 = 0.3662
\]
\[
0.1129 - 0.1093 = 0.0036
\]

### Step 4: Calculate the fraction.
\[
\frac{0.0036}{0.3662} = 0.00983
\]

### Step 5: Multiply by the angle difference.
\[
0.1655 \times 0.00983 = 0.00163
\]

### Step 6: Add to 0.1093.
\[
0.1093 + 0.00163 = 0.11093
\]

### Final Answer:
\[
\boxed{0.11093}
\]

Let me know if you'd like any more details or clarifications.

---

Here are five related questions to expand on this topic:

1. How do you convert degrees, minutes, and seconds to decimal degrees?
2. Can you convert decimal degrees back into degrees, minutes, and seconds?
3. How do we apply trigonometric functions to angles given in degrees, minutes, and seconds?
4. How does linear interpolation work when applied to angles?
5. Can you show more examples of solving mixed angle and decimal problems?

**Tip**: Always ensure you properly convert angles into a consistent format (like decimal degrees) when performing arithmetic operations.

0.1093 + (16° 43' 38.44"- 16° 33' 43") x ((0.1129-0.1093)/(16° 55' 41"-16° 33' 43"))

Solve the expression 0.1093 + (16° 43' 38.44" - 16° 33' 43") x ((0.1129-0.1093)/(16° 55' 41"-16° 33' 43")) step-by-step. This solution includes angle conversion, linear interpolation, and decimal arithmetic, suitable for Grades 10-12.

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