If `tan x^(@) = (5)/(12), tan y = (3)/(4)` and AB = 48 m. Find the length of CD.

Answer

To solve the problem step by step, we will use the given information about the tangents of angles and the length of AB to find the length of CD.

### Step 1: Understand the Problem
We are given:
- \( \tan x = \frac{5}{12} \)
- \( \tan y = \frac{3}{4} \)
- Length of \( AB = 48 \, \text{m} \)

We need to find the length of \( CD \).

### Step 2: Set Up the Triangle
Let's denote:
- \( CD = P \)
- \( BC = x \)

From the triangle, we can express the tangent functions in terms of \( P \) and \( x \).

### Step 3: Use the Tangent Function for Angle \( x \)
For angle \( x \):
\[ \tan x = \frac{P}{AB + BC} = \frac{P}{48 + x} \]
Given \( \tan x = \frac{5}{12} \), we can set up the equation:
\[ \frac{5}{12} = \frac{P}{48 + x} \]

### Step 4: Rearranging the Equation
Cross-multiplying gives:
\[ 5(48 + x) = 12P \]
Expanding this:
\[ 240 + 5x = 12P \quad \text{(Equation 1)} \]

### Step 5: Use the Tangent Function for Angle \( y \)
For angle \( y \):
\[ \tan y = \frac{P}{x} \]
Given \( \tan y = \frac{3}{4} \), we can set up the equation:
\[ \frac{3}{4} = \frac{P}{x} \]

### Step 6: Rearranging the Equation for Angle \( y \)
Cross-multiplying gives:
\[ 3x = 4P \quad \text{(Equation 2)} \]

### Step 7: Substitute Equation 2 into Equation 1
From Equation 2, we can express \( x \) in terms of \( P \):
\[ x = \frac{4P}{3} \]
Now, substitute \( x \) in Equation 1:
\[ 240 + 5\left(\frac{4P}{3}\right) = 12P \]
This simplifies to:
\[ 240 + \frac{20P}{3} = 12P \]

### Step 8: Clear the Fraction
Multiply the entire equation by 3 to eliminate the fraction:
\[ 720 + 20P = 36P \]

### Step 9: Rearranging to Solve for \( P \)
Rearranging gives:
\[ 720 = 36P - 20P \]
\[ 720 = 16P \]

### Step 10: Solve for \( P \)
Dividing both sides by 16:
\[ P = \frac{720}{16} = 45 \]

### Conclusion
The length of \( CD \) is \( 45 \, \text{m} \).