To calculate the percentage contraction of a rod moving with a velocity of 0.8c (where c is the speed of light) in a direction inclined at 60 degrees to its own length, we can use the concept of length contraction.

Length contraction is a phenomenon predicted by Einstein's theory of special relativity. According to this theory, objects moving at relativistic speeds will appear contracted along the direction of motion when observed from a stationary reference frame.

Let's break down the solution into steps:

Step 1: Determine the velocity of the rod relative to the observer
The given velocity is 0.8c, where c is the speed of light. This means the rod is moving at 0.8 times the speed of light relative to the observer.

Step 2: Calculate the Lorentz factor (γ)
The Lorentz factor is given by the equation γ = 1 / √(1 - (v^2 / c^2)), where v is the velocity of the object and c is the speed of light.

In this case, v = 0.8c. Plugging in the values, we get:
γ = 1 / √(1 - (0.8c)^2 / c^2)
= 1 / √(1 - 0.64)
= 1 / √(0.36)
= 1 / 0.6
= 1.67

Step 3: Calculate the contraction factor (β)
The contraction factor is given by the equation β = √(1 - (1/γ^2))

Plugging in the value of γ, we get:
β = √(1 - (1/1.67^2))
= √(1 - 1/2.7889)
= √(1 - 0.358)
= √(0.642)
= 0.802

Step 4: Calculate the percentage contraction
The percentage contraction is given by the equation percentage contraction = (1 - β) * 100

Plugging in the value of β, we get:
percentage contraction = (1 - 0.802) * 100
= 0.198 * 100
= 19.8%

Therefore, the percentage contraction of the rod moving with a velocity of 0.8c in a direction inclined at 60 degrees to its own length is approximately 19.8%.