A man goes 80 m on an inclined cliff inclined at `45^(@)` from the horizontal. Find the vertical height of the man from the ground.

Answer

To find the vertical height of the man from the ground after climbing 80 meters on an inclined cliff at an angle of \(45^\circ\), we can use trigonometric ratios. Here’s a step-by-step solution:

### Step 1: Understand the Triangle
The man climbs along the hypotenuse of a right triangle (let's call it triangle ABC), where:
- AB is the length of the incline (80 m).
- AC is the vertical height we need to find.
- BC is the horizontal distance from the base to the point directly below the man.

### Step 2: Use the Sine Function
In triangle ABC, we can use the sine function, which relates the angle to the opposite side and the hypotenuse. The sine of an angle is defined as:
\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
For our triangle:
\[ \sin(45^\circ) = \frac{AC}{AB} \]

### Step 3: Substitute Known Values
We know that:
- \(AB = 80 \, m\)
- \(\sin(45^\circ) = \frac{1}{\sqrt{2}}\)

Substituting these values into the equation gives:
\[ \frac{1}{\sqrt{2}} = \frac{AC}{80} \]

### Step 4: Solve for AC
To find AC, we can rearrange the equation:
\[ AC = 80 \cdot \sin(45^\circ) \]
Substituting the value of \(\sin(45^\circ)\):
\[ AC = 80 \cdot \frac{1}{\sqrt{2}} = \frac{80}{\sqrt{2}} \]

### Step 5: Simplify AC
To simplify \(\frac{80}{\sqrt{2}}\), we can multiply the numerator and denominator by \(\sqrt{2}\):
\[ AC = \frac{80 \cdot \sqrt{2}}{2} = 40\sqrt{2} \]

### Conclusion
Thus, the vertical height of the man from the ground is:
\[ AC = 40\sqrt{2} \, m \]