Step 1 :Since the distance $s$ varies directly with the square of the time $t$, we can write the equation as $s = kt^2$, where $k$ is a constant.
Step 2 :Given that an object falls 16 feet in one second, we can plug in the values to find the constant $k$: $16 = k(1)^2$, so $k = 16$.
Step 3 :Now we have the equation $s = 16t^2$.
Step 4 :To find the time it takes for the object to fall 80 feet, we set $s$ to 80: $80 = 16t^2$.
Step 5 :Divide both sides by 16: $5 = t^2$.
Step 6 :Take the square root of both sides: $t = \sqrt{5}$.
Step 7 :Round the answer to two decimal places: $t \approx \boxed{2.24}$.