A fence of height H is D feet away from a vertical wall. At what angle θ should a ladder be leaned against the fence in order that the minimum length ladder be required to stretch from the ground to the wall?   Attached is my work so far.

Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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A fence of height H is D feet away from a vertical wall. At what angle θ should a ladder be leaned against the fence in order that the minimum length ladder be required to stretch from the ground to the wall?

 

Attached is my work so far. 

### Problem 2: Trigonometric Analysis and Derivation

#### Diagram Analysis:

The image presents a right triangle with the following labeled parts:
- **Vertical side (opposite side) = y**
- **Horizontal side (adjacent side) = x - D**
- **Hypotenuse = z**
- **Angle at the base = θ**

From the diagram:
- The top vertex of the triangle is labeled as "Wall."
- There is an indication of "H" for the height at "Wall."

#### Equations Derived:

1. **Basic trigonometric identity:**
   \[ 
   \tan(\theta) = \frac{y}{x-D} 
   \]

2. **Solving for y:**
   \[
   y = \frac{Hx}{x-D}
   \]

3. **Expression involving the hypotenuse (z):**
   - Pythagorean theorem:
   \[
   y^2 + x^2 = z^2 
   \]

4. **Substitute y from the first equation:**
   \[
   \left(\frac{Hx}{x-D}\right)^2 + x^2 = z^2
   \]

5. **Solve for z:**
   \[
   z = \left(\left(\frac{Hx}{x-D}\right)^2 + x^2\right)^{\frac{1}{2}}
   \]

#### Explanation:

The derivation above involves the application of trigonometric identities and the Pythagorean theorem. The key focus is solving for \( y \) and expressing \( z \) in terms of \( H \), \( x \), and \( D \). This exercise demonstrates the manipulation of algebraic expressions and basic trigonometric principles to analyze parameters for a given geometric structure (right triangle in this case).
Transcribed Image Text:### Problem 2: Trigonometric Analysis and Derivation #### Diagram Analysis: The image presents a right triangle with the following labeled parts: - **Vertical side (opposite side) = y** - **Horizontal side (adjacent side) = x - D** - **Hypotenuse = z** - **Angle at the base = θ** From the diagram: - The top vertex of the triangle is labeled as "Wall." - There is an indication of "H" for the height at "Wall." #### Equations Derived: 1. **Basic trigonometric identity:** \[ \tan(\theta) = \frac{y}{x-D} \] 2. **Solving for y:** \[ y = \frac{Hx}{x-D} \] 3. **Expression involving the hypotenuse (z):** - Pythagorean theorem: \[ y^2 + x^2 = z^2 \] 4. **Substitute y from the first equation:** \[ \left(\frac{Hx}{x-D}\right)^2 + x^2 = z^2 \] 5. **Solve for z:** \[ z = \left(\left(\frac{Hx}{x-D}\right)^2 + x^2\right)^{\frac{1}{2}} \] #### Explanation: The derivation above involves the application of trigonometric identities and the Pythagorean theorem. The key focus is solving for \( y \) and expressing \( z \) in terms of \( H \), \( x \), and \( D \). This exercise demonstrates the manipulation of algebraic expressions and basic trigonometric principles to analyze parameters for a given geometric structure (right triangle in this case).
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