https://file.aiquickdraw.com/mathbot/file/17273677053573okehod5.pdf

Let's go through the questions one by one and provide solutions based on the uploaded assignment.

### **Question 1: Solve the triangles.**

This involves solving two triangles. Since there are no specific values mentioned in the PDF, you need to use the Law of Sines or the Law of Cosines, depending on the information given in the triangles. 

#### a) General steps for solving a triangle:
1. **If two angles and one side are known (AAS or ASA):**
   - Use the **Law of Sines**: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).
   - Find missing sides and angles.

2. **If two sides and one angle (SAS or SSA) are given:**
   - Use **Law of Cosines** to find missing angles or sides:
   \[
   c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
   \]
   - Then use Law of Sines to find remaining unknowns.

Since no specific triangle is mentioned in the file, let me know if you need help with a specific problem or triangle.

---

### **Question 2: A triangular lot with sides 180 m, 160 m, and 123.5 m. Find the angles.**

To find the angles of the triangle, we use the **Law of Cosines**. For a triangle with sides \(a = 180\), \(b = 160\), and \(c = 123.5\), the angles are calculated as:

\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]
\[
\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}
\]
\[
\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}
\]

Substitute the values and compute the angles.

---

### **Question 3: Hill with a 35° grade cut down to 10° grade, 800 m distance.**

1. **Original height of the hill**: 
   \[
   h_1 = 800 \cdot \sin(35°)
   \]

2. **New height after the cut**: 
   \[
   h_2 = 800 \cdot \sin(10°)
   \]

3. **Vertical cut removed**: 
   \[
   \text{Vertical cut} = h_1 - h_2
   \]

4. **New distance from bottom to top for the roadbed**: 
   Use the Pythagorean theorem if necessary after calculating the new height.

---

### **Question 4: Guy wire forms a 42° angle, 300 N pull.**

1. **Vertical component**: 
   \[
   F_{\text{vertical}} = 300 \cdot \sin(42°)
   \]

2. **Horizontal component**: 
   \[
   F_{\text{horizontal}} = 300 \cdot \cos(42°)
   \]

---

### **Question 5: Structural member BC length 60 mm, find angle ɸ.**

Without a figure, it's hard to provide exact steps, but typically this involves using trigonometric relationships or the Law of Cosines.

---

### **Question 6: Calculate X and Y components of forces \(P_1 = 7.78 \, N\) and \(P_2 = 10 \, N\)**

1. For each force, decompose into components:
   - \(P_{1x} = 7.78 \cdot \cos(\theta_1)\), \(P_{1y} = 7.78 \cdot \sin(\theta_1)\)
   - \(P_{2x} = 10 \cdot \cos(\theta_2)\), \(P_{2y} = 10 \cdot \sin(\theta_2)\)

2. **Resultant force**:
   \[
   F_{\text{resultant}} = \sqrt{(P_{1x} + P_{2x})^2 + (P_{1y} + P_{2y})^2}
   \]

---

### **Question 7: Crane applies 9500 N, find vertical and horizontal components.**

1. **Vertical component**: 
   \[
   F_{\text{vertical}} = 9500 \cdot \sin(\theta)
   \]

2. **Horizontal component**: 
   \[
   F_{\text{horizontal}} = 9500 \cdot \cos(\theta)
   \]

Let me know if you need help with any specific problem from the assignment or further clarification! 

### 5 Follow-Up Questions:
1. How do you apply the Law of Cosines to solve for unknown angles in a triangle?
2. Can you explain the difference between solving a triangle using Law of Sines and Law of Cosines?
3. How does the angle of a hill's slope affect the vertical height calculation?
4. What are the horizontal and vertical components of a force and how do they relate to the overall force?
5. How can you use trigonometric ratios to find missing angles and sides in a triangle?

### Tip:
When solving triangles, always double-check that the sum of the angles equals 180° as a validation step.

Solve the triangles, calculate angles, and determine the components of forces for various problems related to structural engineering and physics.

This page provides step-by-step solutions to a series of problems related to solving triangles, calculating angles in a triangular lot, and determining vertical and horizontal force components in engineering scenarios. Concepts include the Law of Sines, Law of Cosines, and trigonometric force decomposition. Suitable for engineering and physics students.

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