To find the direct straight-line distance from the man on the fence to the lion on the ground, we can use trigonometry, specifically the tangent function and properties of right triangles.
- We know the height of the fence (which is also the height of the man observing) is 4.2 meters.
- The angle of depression to the lion is 24 degrees.
Since the angle of depression is measured from the horizontal line of sight of the person down to the lion, the angle of elevation from the lion's position to the top of the fence is also 24 degrees (because they are alternate interior angles).
Now, we can set up our right triangle:
- The side opposite the 24-degree angle is the height of the fence (4.2 meters).
- We want to find the hypotenuse (the direct distance \( d \) from the man to the lion).
Using the sine function, we have:
\[ \sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}} \implies \sin(24^\circ) = \frac{4.2}{d} \]
Now we can reorganize the equation to solve for \( d \):
\[ d = \frac{4.2}{\sin(24^\circ)} \]
Next, we can calculate \( \sin(24^\circ) \):
\[ \sin(24^\circ) \approx 0.4067 \]
Now substituting this value back into the equation:
\[ d = \frac{4.2}{0.4067} \approx 10.34 \]
Finally, rounding to the nearest tenths place, the direct straight-line distance from the man to the lion is:
\[ \boxed{10.3} \text{ meters} \]