Answer
$4$ seconds
Work Step by Step
Let's note by $t$ the time (in seconds) needed by the cheetah to catch up to the antelope.
The distance, $C$, travelled by the cheetah running at a speed of $90$ feet per second will be $90$ times $t$. Therefore $$C = 90t.\tag{1}$$ Similarly, the distance, $A$, travelled by the antelope running at $60$ feet per second in $t$ seconds will be $60t$. However, the antelope has a headstart of $120$ ft. Hence, the total distance the antelope travelled (from the origin point) is $60t + 120$. Therefore $$A = 60t + 120.\tag{2}$$ In order for the cheetah to catch up, the distances $C$ and $A$ should be equal. Therefore, we set up the equation $$C = A.\tag{3}$$ Using equations $(1)$ and $(2)$ in $(3)$ we have:
$$\begin{align}
90t& = 60t + 120\\
30t& = 120\\
t &= 4.
\end{align}$$ Thus the time taken by cheetah to catch up is $4$ seconds.
